Download 1 1. A uniform lamina L is formed by taking a uniform square sheet of

Level N Mechanics Top 60 questions
1.
10 cm
A
B
10 cm
L
D
C
A uniform lamina L is formed by taking a uniform square sheet of material ABCD, of side 10
cm, and removing the semi–circle with diameter AB from the square, as shown in the figure
above. The center of mass of a uniform semi–circular lamina, radius a, is at a
4a
distance
from the center of the bounding diameter. Use π 3 .
3π
(a)
The distance of the center of mass of the lamina L from the midpoint of AB, in cm as
a fraction in lowest terms, is
The lamina is freely suspended from D and hangs at rest. Assume that the distance of the
center of mass of the lamina L from the midpoint of AB is 6 cm.
(b)
is
2.
The angle between CD and the vertical is α. The value of tan α, to one decimal place,
A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion
of the lorry has magnitude 750 N and the lorry’s engine is working at a rate of 36 kW. Use g =
10 m/s2.
(a)
is
The acceleration of the lorry when its speed is 20 m/s, in m/s2 to one decimal place,
The lorry comes to a hill inclined at an angle α to the horizontal, where sin α =
1
.
10
The
magnitude of the resistance to motion from non–gravitational forces remains 750 N. The lorry
moves up the hill at a constant speed of 20 m/s.
(b)
The rate at which the lorry’s engine is now working, to the nearest kW, is
1
Level N Mechanics Top 60 questions
3.
Two small spheres P and Q of equal radius have masses m and 5m respectively. They lie on a
smooth horizontal table. Sphere P is moving with speed u when it collides directly with sphere
1
5
Q which is at rest. The coefficient of restitution between the spheres is e, where e > .
(a)
The speed of P immediately after the collision is in the form
u
( ae − b ) . The values
6
of a and b, to the nearest whole numbers, respectively are
Three small spheres A, B and C of equal radius lie at rest in a straight line on a smooth
horizontal table, with B between A and C. A and C each have mass 5m, and the mass of B is m.
B is projected towards C with speed u. The coefficient of restitution between each pair of
spheres is 4 .
5
(b)
The speed of B after colliding with C is in the form nu. The value of n, to one
decimal place, is
(c)
Assume the speed of B after colliding with C is 2u and that its direction of motion is
reversed. B then collides with A. The speed of B after colliding with A is in the form ku. The
value of k, to the nearest whole number, is
4.
A particle P of mass 1 kg is moving under the action of a single force F newtons. At time t
⎛3
⎞
seconds the velocity of P, v m/s, is given by v = ⎜ t 2 − 2 ⎟ i + ( 4t ) j .
⎝2
⎠
(a)
The magnitude of F when t = 1, to the nearest N, is
When t = 0, P is at the point A. The position vector of A with respect to a fixed origin O
is ( i − 2 j) m . When t = 2, P is at the point B.
(b)
The distance of B from O is d m. The value of d2, to the nearest whole number, is
5.
A van of mass 1500 kg is driving up a straight road inclined at an angle α to the horizontal,
1
6
where sin α = . The resistance to motion due to non–gravitational forces is modeled as a
constant force of magnitude 1000 N. It is given that the initial speed of the van is 30 m/s and
that the van’s engine is working at a rate of 60 kW. Use g = 10 m/s2.
(a)
The initial deceleration of the van, to the nearest m/s2, is
When traveling up the same hill, the rate of working of the van’s engine is increased to 70
kW. Use the model for the resistance due to non–gravitational forces.
2
Level N Mechanics Top 60 questions
(b)
The constant speed which can be sustained by the van at this rate of working, to the
nearest m/s, is
(c)
Which of the following is a valid reason why the use of this model for resistance
may mean that the answer to the previous question is too high?
6.
Two particles A and B move on a smooth horizontal table. The mass of A is m, and the mass of
B is 4m. Initially A is moving with speed u when it collides directly with B, which is at rest on
the table. As a result of the collision, the direction of motion of A is reversed. The coefficient
of friction between the particles is e.
In the subsequent motion, B strikes a smooth vertical wall and rebounds. The wall is
perpendicular to the direction of motion of B. The coefficient of restitution between B and the
wall is 4 . There is a second collision between B and A.
5
(a)
The range of values of e is p < e < q. The values of p and q, as fractions in lowest
terms, respectively are
(b)
Assume
that
the
speeds
of
A
and
B
after
the
first
collision
u
u
( 3e + 1) and ( 6e − 1) respectively. Given that e = 13 , the total kinetic energy lost in the
5
5
first collision is in the form kmu2. The value of k, to two decimal places, is
are
A
7.
6 cm
P
B
M
S
C
3 cm
Q
R
2 cm
4 cm
3
Level N Mechanics Top 60 questions
The figure above shows a decoration which is made by cutting the shape of a simple tree from
a sheet of uniform card. The decoration consists of a triangle ABC and a rectangle PQRS. The
points P and S lie on BC and M is the midpoint of both BC and PS. The triangle ABC is
isosceles with AB = AC, BC = 4 cm, AM = 6 cm, PS = 2 cm and PQ = 3 cm.
(a)
The distance of the center of mass of the decoration from BC, in cm as a fraction in
lowest terms, is
The decoration is suspended from Q and hangs freely.
(b)
Assume that the distance of the center of mass of the decoration from BC is 1 cm.
The angle between PQ and the vertical is denoted by α. The value of tan α, as a fraction in
lowest terms, is
8.
•
C
v0
A
1m
T
3m
θ
xm
The object of a game is to throw a ball B from a point A to hit a target T which is placed at the
top of a vertical pole, as shown in the figure above. The point A is 1 m above horizontal
ground and the height of the pole is 3 m. The pole is at a horizontal distance of x = 16 m from
A. The ball B is projected from A with speed v0 = 20 m/s at an angle of elevation of θ = 37°. B
hits the pole at the point C. B and T are modeled as particles. Use g = 10 m/s2.
1
⎛
⎞
D
D
D
D
⎜ sin 37 0.6, cos 37 0.8, cos 60 = , tan 60 = 3 1.7 ⎟
2
⎝
⎠
(a)
The time taken for B to move from A to C, to the nearest s, is
(b)
Assume that the time taken for B to move from A to C is 0.1 s. The distance CT, to
the nearest cm, is
B is thrown again from A which is now at x = 10 m from the pole. The speed of projection of
B is changed to V m/s and the angle of elevation is set at θ = 60°. This time B hits T.
4
Level N Mechanics Top 60 questions
(c)
The value of V is in the form
p
m/s . The value of p, to the nearest whole number,
3
is
(d)
9.
Explain why, in practice, a range of values of V results in B hitting T.
A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The
tow–bar joining the car to the trailer is modeled as a light rod parallel to the road. The total
resistance to motion of the car is modeled as having constant magnitude 750 N. The total
resistance to motion of the trailer is modeled as of magnitude R newtons, where R is a
constant. When the engine of the car is working at a rate of 50 kW, the car and the trailer
travel at a constant speed of 25 m/s.
(a)
The value of R, to the nearest N, is
When traveling at 25 m/s the driver of the car disengages the engine and applies the brakes.
The brakes provide a constant braking force of magnitude 250 N on the car. The resisting
force to motion of the car remains 750 N. Assume that the resisting force to motion of the
trailer is 250 N.
(b)
The deceleration of the car while braking, in m/s2 to one decimal place, is (Consider
the system consisting of car and trailer.)
(c)
Assume that the deceleration of the car is 0.2 m/s2. The thrust in the tow–bar while
braking, to the nearest N, is (Consider the system consisting of car alone.)
(d)
Assume that the deceleration of the car is 0.625 m/s2. The absolute value of the work
done by the braking force in bringing the car and the trailer to rest, to the nearest kJ, is (Find
the distance covered by the car before coming to rest first.)
5
Level N Mechanics Top 60 questions
10.
20 m/s
α
A
B
17 m
7m
O
C
A particle P is projected from a point A with speed 20 m/s at an angle of elevation α, where
3
5
sin α = . The point O is on horizontal ground, with O vertically below A and OA = 17 m. P
moves freely under gravity and passes through a point B, which is 7 m above the ground,
before reaching the ground at the point C, as shown in the figure above. Use g = 10 m/s2.
⎛
1
3 ⎞
D
D
D
D
,⎟
⎜ sin 30 = cos 60 = ,sin 60 = cos 30 =
2
2 ⎟
⎜
⎜ sin 37D = cos 53D 0.6,sin 53D = cos 37D 0.8 ⎟
⎜
⎟
2
⎜
⎟
D
D
⎜ sin 45 = cos 45 = 2
⎟
⎝
⎠
(a)
The time of flight from A to C, in s to one decimal place, is
(b)
is
Assume that the time of flight from A to C is 2 s. The distance OC, to the nearest m,
(c)
The square of the speed of P at B, to the nearest whole number, is
(d)
Assume that vx = 15 m/s and vy = 20 m/s at B. The angle that the velocity of P at B
makes with the horizontal, to the nearest degree, is
6
Level N Mechanics Top 60 questions
B
11.
2a
C
1
a
2
D
θ
A
A uniform ladder of weight W and length 2a, rests in equilibrium with one end A on a smooth
horizontal floor and the other end B on a rough vertical wall. The ladder is in a vertical plane
perpendicular to the wall. The coefficient of friction between the wall and the ladder is µ. The
ladder makes an angle θ with the floor, where tan θ = 2. A horizontal light inextensible string
1
2
CD is attached to the ladder at the point C, where AC = a . The string is attached to the wall
1
4
at the point D, with BD vertical, as shown in the figure above. The tension in the string is W .
The ladder is modeled as a rod.
(a)
The magnitude of the force of the floor on the ladder is in the form pW. The value of
p, as a fraction in lowest terms, is
(b)
Assume the magnitude of the force of the floor on the ladder is
15
W . The minimum
16
value of µ, as a fraction in lowest terms, is
12. A smooth sphere P of mass m is moving in a straight line with speed u on a smooth horizontal
table. Another smooth sphere Q of mass 2m is at rest on the table. The sphere P collides
directly with Q. After the collision the direction of motion of P is unchanged. The spheres
have the same radii and the coefficient of restitution between P and Q is e. The spheres are
modeled as particles.
(a)
The maximum value of e, as a fraction in lowest terms, is
(b)
Given that e = , the loss of kinetic energy in the collision is kmu2. The value of k, as
1
4
a fraction in lowest terms, is
7
Level N Mechanics Top 60 questions
(c)
The lost kinetic energy may have been transformed to which of the following
energies?
1.
2.
3.
4.
5.
Chemical potential
Gravitational potential
Heat
Nuclear
Sound
13. A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity
of the ball is ( −30i ) m/s . Immediately after being struck, the velocity of the ball
is (16i + 20 j) m/s .
(a)
The square of the magnitude of the impulse exerted on the ball by the bat, to the
nearest whole number, is
In the subsequent motion, the position vector of the ball is r meters at time t seconds. In a
model of the situation, it is assumed that r = ⎡⎣16ti + ( 20t − 5t 2 ) j⎤⎦ .
(b)
14.
The square of the speed of the ball when t = 3, to the nearest whole number, is
y
A•
4m
4
O
9
B
•
x
6m
4
C • 2m
8
Level N Mechanics Top 60 questions
The figure above shows a triangular lamina ABC. The coordinates of A, B and C are (0, 4), (9,
0) and (0, –4) respectively. Particles of mass 4m, 6m and 2m are attached at A, B and C
respectively.
(a)
The x–coordinate of the center of mass of the three particles, without the lamina, to
one decimal place, is
(b)
The y–coordinate of the center of mass of the three particles, without the lamina, as a
fraction in lowest terns, is
15. A uniform ladder AB, of mass m and length 2a, has one end A on rough horizontal ground. The
coefficient of friction between the ladder and the ground is 0.6. The other end B of the ladder
rests against a smooth vertical wall. A builder of mass 10m stands at the top of the ladder. To
prevent the ladder from slipping, the builder’s friend pushes the bottom of the ladder
horizontally towards the wall with a force of magnitude P. This force acts in a direction
perpendicular to the wall. The ladder rests in equilibrium in a vertical plane perpendicular to
3
2
the wall and makes an angle α with the horizontal, where tan α = .
(a)
The reaction of the wall on the ladder has magnitude kmg. The value of k, to the
nearest whole number, is
(b)
Assume that the reaction of the wall on the ladder has magnitude 10mg. The range of
values of P for which the ladder remains in equilibrium is Pmin ≤ P ≤ Pmax . The sum Pmin + Pmax ,
to the nearest whole number, is
16.
x
B
•
A
•
θ
A particle P of mass 1 kg is projected from a point A up a line of greatest slope AB of a fixed
plane. The plane is inclined at an angle θ = 37° to the horizontal and AB = x = 3 m with B
above A, a shown in the figure above. The speed of P at A is 10 m/s.
Use g = 10 m/s2, sin 37° = 0.6, cos 37° = 0.8.
Assume the plane is smooth.
(a)
The speed of P at B, to the nearest m/s, is
The plane is now assumed to be rough. At A, the speed of P is 10 m/s and at B the speed of P
is 49.6 m/s .
(b)
The coefficient of friction between P and the plane, to one decimal place, is
9
Level N Mechanics Top 60 questions
17. A particle P moves in a horizontal plane. At time t seconds, the position vector of P is r meters
relative to a fixed origin O, and r is given by r = (18t − 4t 3 ) i + ct 2 j , where c is a positive
constant. When t = 1.5, the speed of P is 15 m/s.
(a)
The value of c, to the nearest whole number, is
Assume c = 2 in the previous question.
(b)
18.
The magnitude of the acceleration of P at t =
1
8
s , to the nearest m/s2, is
A
10 m/s
B
30 m
α
10 m
C
In a ski–jump competition, a skier of mass 80 kg moves from rest at a point A on a ski–slope.
The skier’s path is an arc AB. The starting point A of the slope is 30 m above horizontal
ground. The end B of the slope is 10 m above the ground. When the skier reaches B, she is
traveling at 10 m/s, and moving upwards at an angle α = 30° to the horizontal, as shown in the
figure above. The distance along the slope from A to B is 100 m. The resistance to motion
while she is on the slope is modeled as a force of constant magnitude R N.
1
3
.
Use g = 10 m/s2, sin 30D = , cos 30D =
2
2
(a)
The value of R, to the nearest N, is
On reaching B, the skier then moves through the air and reaches the ground at the point C. The
motion of the skier in moving from B to C is modeled as that of a particle moving freely under
gravity.
(b)
The time for the skier to move from B to C, to the nearest s, is
(c)
Assume that the time for the skier to move from B to C is 3 s .The horizontal distance
from B to C, to the nearest m. is
10
Level N Mechanics Top 60 questions
(d)
In this question use the equation of vy vs. y (not vs. t). The speed of the skier
immediately before she reaches C is v. The value of v2, to the nearest whole number, is
A
19.
B
X
a
2a
C
D
a
F
E
2a
A uniform lamina ABCDEF is formed by taking a uniform sheet of card in the form of a
square AXEF, of side 2a, and removing the square BXDC of side a, where B and D are the
midpoints of AX and XE respectively, as shown in the figure above.
(a)
The distance of the center of mass of the lamina from AF is ka. The value of k, as a
fraction in lowest terms, is
The lamina is freely suspended from A and hangs in equilibrium. Assume that the distance of
6
7
the center of mass of the lamina from AF is a .
(b)
The angle which AF makes with the vertical, to the nearest degree, is
⎛
1
3 ⎞
D
D
D
D
,⎟
⎜ sin 30 = cos 60 = ,sin 60 = cos 30 =
2
2 ⎟
⎜
⎜ sin 37D = cos 53D 0.6,sin 53D = cos 37D 0.8 ⎟
⎜
⎟
2
⎜
⎟
D
D
⎜ sin 45 = cos 45 = 2
⎟
⎝
⎠
20. Two small spheres A and B have mass 3m and 2m respectively. They are moving towards each
other in opposite directions on a smooth horizontal plane, both with speed 2u, when they
collide directly. As a result of the collision, the direction of motion of B is reversed and its
speed is unchanged.
(a)
The coefficient of restitution between the spheres, as a fraction in lowest terms, is
(b)
Subsequently, B collides directly with another small sphere C of mass 5m which is at
rest. The coefficient of restitution between B and C is 3 . After B collides with C, will there be
5
there be a second collision between A and B?
11
Level N Mechanics Top 60 questions
21. A particle P moves in a straight line so that, at time t seconds, its acceleration a m/s2 is given
by
a = 4t − t 2 , 0 ≤ t ≤ 3.
27
a = 2 , t > 3.
t
At t = 0, P is at rest.
The speed of P when t = 3, to the nearest m/s, is
(a)
(b)
Assume the speed of P at t = 3, is 3 m/s. The speed of P at t = 6, in m/s to one decimal
place, is
22.
24 cm
A
8 cm
O
X
B
The figure above shows a template T made by removing a circular disc, of center X and radius
8 cm, from a uniform circular lamina, of center O and radius 24 cm. The point X lies on the
diameter AOB of the lamina and AX = 16 cm. The center of mass of T is at the point G.
(a)
The value of AG, to the nearest cm, is
The template T is free to rotate about a smooth fixed horizontal axis, perpendicular to the
1
4
plane of T, which passes through the midpoint of OB. A small stud of mass m is fixed at B,
and T and the stud are in equilibrium with AB horizontal. The stud is modeled as a particle.
The mass of T is M. Assume that AG = 30 cm.
(b)
M = pm; the value of p, as a fraction in lowest terms, is
12
Level N Mechanics Top 60 questions
23. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and
4
3
is inclined at an angle θ to the horizontal where tan θ = . The distance traveled down the
chute by each brick is 10 m. A brick of mass 1 kg is released from rest at the top of the chute.
When it reaches the bottom of the chute, its speed is 4 m/s. Use g = 10 m/s2.
(a)
The coefficient of friction between the brick and the chute, to one decimal place, is
Another brick of mass 1 kg slides down the chute. This brick is given an initial speed
of 2 11 m/s at the top of the chute. Assume the coefficient of friction between the brick and
the chute is 0.5.
(b)
The speed of the brick when it reaches the bottom of the chute, to the nearest m/s, is
24.
A
W 2a
12a
N•
•
O
X
B
Z
C
2a
Y
8a
The figure above shows a template made by removing a square WXYZ from a uniform
triangular lamina ABC. The lamina is isosceles with CA = CB and AB = 12a. The midpoint of
AB is N and NC = 8a. The center O of the square lies on NC and ON = 2a. The sides WX and
ZY are parallel to AB and WZ = 2a. The center of mass of the template is at G.
(a)
NG = pa; the value of p, as a fraction in lowest terms, is
The template has mass M. A small metal stud of mass kM is attached to the template at C. The
center of mass of the combined template and stud lies on YZ. The stud is modeled as a
particle. Assume that NG = 2.6a.
(b)
k can be written in the form
n
100
. The value of n, to the nearest whole number, is
13
Level N Mechanics Top 60 questions
25.
D
θ
A
C
B
0.5 m
A uniform beam of mass 1.6 kg is freely hinged at one end A to a vertical wall. The beam is
held in equilibrium in a horizontal position by a rope which is attached to a point C on the
beam, where AC = 0.5 m. The rope is attached to the point D on the wall vertically above A,
4
3
where ∠ACD = θ = tan −1 , as shown in the figure above. The beam is modeled as a uniform
rod and the rope as a light inextensible string. The tension in the rope is 60 N. Use g = 10
m/s2.
(a)
The length of AB, in m to one decimal place, is
(b)
H and V are the magnitudes of the horizontal and the vertical components
respectively of the resultant reaction of the hinge on the beam at A. The ratio H to V, as a
fraction in lowest terms, is
26. A particle P of mass m is moving in a straight line on a smooth horizontal table. Another
particle Q of mass km is at rest on the table. P collides directly with Q. The direction of motion
of P is reversed by the collision. After the collision, the speed of P is v and the speed of Q is
3v. The coefficient of restitution between P and Q is 1 .
2
(a)
The value of k, to the nearest whole number, is
After being struck by P, Q collides directly with a particle R of mass 11m which is at rest on
the table. After this second collision, Q and R have the same speed and are moving in opposite
directions. Assume k = 2.
(b)
The coefficient of restitution between Q and R, as a fraction in lowest terms, is
14
Level N Mechanics Top 60 questions
27. A dart player throws darts at a dart board which hangs vertically. The motion of the dart is
modeled as that of a particle moving freely under gravity. The darts move in a vertical plane
which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed
10 m/s. It hits the board at a point which is 45 cm below the level from which it was thrown.
Use g = 10 m/s2. ( sin 37D 0.6, cos 37D 0.8 )
(a)
The horizontal distance from the point where the dart was thrown to the dart board,
to the nearest m, is
The dart player moves his position. He now throws a dart from a point which is at a horizontal
distance of 9.6 m from the board. He throws the dart at an angle of elevation of 37°. This dart
hits the board at a point which is at the same level as the point from which it was thrown.
(b)
28.
The speed with which the dart is thrown, to the nearest m/s, is
A 4m
O
•
m B
•
2a
D
5a
•C
2m
A loaded plate L is modeled as a uniform rectangular lamina ABCD and three particles. The
sides CD and AD of the lamina have lengths 5a and 2a respectively and the mass of the
lamina is 3m. The three particles have mass 4m, m and 2m and are attached at the points A, B
and C respectively, as shown in the figure above.
(a)
The distance of the center of mass of L from AD is pa. The value of p, as a fraction
in lowest terms, is
15
Level N Mechanics Top 60 questions
(b)
The distance of the center of mass of L from AB is qa. The value of q, to one decimal
place, is
The point O is the midpoint of AB. The loaded plate L is freely suspended from O and hangs
at rest under gravity.
Assume that the distance of the center of mass of L from AD is 2.2a and the distance of the
center of mass of L from AB is 0.75a.
(c)
The angle that AB makes with the horizontal is α. The value of tan α, to one decimal
place, is
A horizontal force of magnitude P is applied at C in the direction CD. The loaded plate L
remains suspended from O and rests in equilibrium with AB horizontal and C vertically below
B.
Assume that the distance of the center of mass of L from AD is 2.2a.
(d)
P = kmg; the value of k, to one decimal place, is
Assume that the force P is 2 11 mg .
(e)
The magnitude of the force on L at O is nmg. The value of n, to the nearest whole
number, is
29.
D
A
θ
•
C
B
x
2a
The figure above shows a horizontal uniform pole AB, of weight W and length 2a. The end A
of the pole rests on a rough vertical wall. One end of a light inextensible string BD is attached
to the pole at B and the other end is attached to the wall at D. A particle of weight 2W is
attached to the pole at C, where BC = x. The pole is in equilibrium in a vertical plane
perpendicular to the wall. The string BD is inclined at an angle θ to the horizontal,
3
5
where sin θ = . The pole is modeled as a uniform rod.
16
Level N Mechanics Top 60 questions
(a)
The tension in BD is in the form
5 ( pa − qx )
W . The ratio p to q, to one decimal
6a
place, is
7
5
The vertical component of the force exerted by the wall on the pole is W . Assume p = 4 and
q = 1 in the expression of the tension.
x can be written in the form x = ka . The value of k, to one decimal place, is
(b)
Assume k = 1 in the expression of x in terms of a.
(c)
The horizontal component of the force exerted by the wall on the pole can be written
in the form H = nW . The value of n, to one decimal place, is
30.
u m/s
θ
A
C
45 m
30 m
B
D
A particle P is projected from a point A with speed u m/s at an angle of elevation θ,
4
5
where cos θ = . The point B, on horizontal ground, is vertically below A and AB = 45 m.
After projection, P moves freely under gravity passing through a point C, 30 m above the
ground, before striking the ground at the point D, as shown in the figure above.
It is given that P passes through C with speed 20 m/s. Use g = 10 m/s2.
(a)
The value of u, to the nearest m/s, is
(b)
Assume that u = 7.5 m/s. The angle which the velocity of P makes with the
horizontal as P passes through C is α. The value of cos α, to one decimal place, is
(c)
(
)
Assume that u = 25 m/s. The distance BD can be written in the form k 1 + 5 m .
The value of k, to the nearest whole number, is
17
Level N Mechanics Top 60 questions
31.
Figure 2
A
A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened
at one end to a point A. The other end of the spring is fastened to a small wooden block B of
mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modeled as a
particle.
The table is initially assumed to be smooth. The block is released from rest when it is a
distance 0.3 m from A. By using the principle of the conservation of energy,
(a)
find, to 3 significant figures, the speed of B when it is a distance 0.25 m from A.
It is now assumed that the table is rough and the coefficient of friction between B and the
table is 0.6.
(b)
Find, to 3 significant figures, the minimum distance from A at which B can rest in
equilibrium.
32. A particle P moves in a straight line with simple harmonic motion about a fixed centre O with
period 2 s. At time t seconds the speed of P is v m s−1. When t = 0, v = 0 and P is at a point A
where OA = 0.25 m.
Find the smallest positive value of t for which AP = 0.375 m.
18
Level N Mechanics Top 60 questions
33. Figure 3
An ornament S is formed by removing a solid right circular cone, of radius r and height
1
h,
2
from a solid uniform cylinder, of radius r and height h, as shown in Fig. 3.
(a)
Show that the distance of the centre of mass S from its plane face is
17
h.
40
The ornament is suspended from a point on the circular rim of its open end. It hangs in
equilibrium with its axis of symmetry inclined at an angle α to the horizontal. Given that
h = 4r,
(b)
find, in degrees to one decimal place, the value of α.
34. A light elastic string AB of natural length 1.5 m has modulus of elasticity 20 N. The end A is
fixed to a point on a smooth horizontal table. A small ball S of mass 0.2 kg is attached to the
end B. Initially S is at rest on the table with AB = 1.5 m. The ball S is then projected
horizontally directly away from A with a speed of 5 m s−1. By modelling S as a particle,
(a)
find the speed of S when AS = 2 m.
When the speed of S is 1.5 m s−1, the string breaks.
(b)
Find the tension in the string immediately before the string breaks.
19
Level N Mechanics Top 60 questions
35. Figure
A
A small ring R of mass in is free to slide on a smooth straight wire which is fixed at an angle
of 30° to the horizontal. The ring is attached to one end of a light elastic string of natural
length a and modulus of elasticity λ. The other end of the string is attached to a fixed point A
9
of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point B, where AB = a .
8
(a)
Show that λ = 4mg.
1
a , and released from rest. At time t
4
⎛1
⎞
after R is released the extension of the string is ⎜ a + x ⎟ .
⎝8
⎠
The ring is pulled down to the point C, where BC =
(b)
Obtain a differential equation for the motion of R while the string remains taut, and
⎛a⎞
show that it represents simple harmonic motion with period π ⎜ ⎟ .
⎝g⎠
(c)
Find, in terms of g, the greatest magnitude of the acceleration of R while the string
remains taut.
(d)
Find, in terms of a and g, the time taken for R to move from the point at which it first
reaches maximum speed to the point where the string becomes slack for the first time.
20
Level N Mechanics Top 60 questions
36.
A child’s toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The
plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig.
The cylinder and the hemisphere each have radius r, and the height of the cylinder is h. The
material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in
equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the
cylinder vertical.
(a)
Show that the distance d of the centre of mass of the toy from its lowest point O is
given by
d=
h 2 + 2hr + 5r 2
.
2(h + 4r )
When the toy is placed with any point of the curved surface of the hemisphere resting on the
plane it will remain in equilibrium.
(b)
Find h in terms of r.
21
Level N Mechanics Top 60 questions
37.
A
Part of a hollow spherical shell, centre O and radius a, is removed to form a bowl with a plane
circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point A is
the lowest point of the bowl. The point B is on the rim of the bowl and ∠AOB = 120°, as
shown in Fig. 4. A smooth small marble of mass m is placed inside the bowl at A and given an
initial horizontal speed u. The direction of motion of the marble lies in the vertical plane AOB.
The marble stays in contact with the bowl until it reaches B. When the marble reaches B, its
speed is v.
(a)
Find an expression for v2.
(b)
For the case when u2 = 6ga, find the normal reaction of the bowl on the marble as the
marble reaches B.
(c)
Find the least possible value of u for the marble to reach B.
The point C is the other point on the rim of the bowl lying in the vertical plane OAB.
(d)
Find the value of u which will enable the marble to leave the bowl at B and meet it
again at the point C.
22
Level N Mechanics Top 60 questions
38. A rough disc rotates in a horizontal plane with constant angular velocity ω about a fixed
4
vertical axis. A particle P of mass m lies on the disc at a distance a from the axis. The
3
3
coefficient of friction between P and the disc is . Given that P remains at rest relative to the
5
disc,
9g
.
(a)
prove that ω 2 ≤
20a
The particle is now connected to the axis by a horizontal light elastic string of natural length a
and modulus of elasticity 2mg. The disc again rotates with constant angular velocity ω about
4
the axis and P remains at rest relative to the disc at a distance a from the axis.
3
2
(b)
Find the greatest and least possible values of ω .
39. Figure 4
A
B
A particle P of mass m is attached to two light inextensible strings. The other ends of the
string are attached to fixed points A and B. The point A is a distance h vertically above B. The
system rotates about the line AB with constant angular speed ω. Both strings are taut and
inclined at 60° to AB, as shown in Fig. 4. The particle moves in a circle of radius r.
(a)
(b)
3
h.
2
Find, in terms of m, g, h and ω, the tension in AP and the tension in BP.
Show that r =
The time taken for P to complete one circle is T.
(c)
⎛ 2h ⎞
Show that T < π ⎜ ⎟ .
⎝ g ⎠
23
Level N Mechanics Top 60 questions
40. Figure 2
A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the
same material. The centre O of the base of the cone is also the centre of one end of the
cylinder, as shown in Fig. 2. The radius of the cylinder is r and the radius of the base of the
cone is 2r. The height of the cone and the height of the cylinder are each h. The centre of mass
of the model is at the point G.
(a)
Show that OG =
1
h.
14
(8)
The model stands on a desk top with its plane face in contact with the desk top. The desk top
7
. The desk top is rough
is tilted until it makes an angle α with the horizontal, where tan α =
26
enough to prevent slipping and the model is about to topple.
(b)
Find r in terms of h.
24
Level N Mechanics Top 60 questions
41. A particle P of mass m kg slides from rest down a smooth plane inclined at 30° to the
horizontal. When P has moved a distance x metres down the plane, the resistance to the
motion of P from non-gravitational forces has magnitude mx2 newtons.
Find
(a)
the speed of P when x = 2,
(b)
the distance P has moved when it comes to rest for the first time.
42. A particle P of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the
spring is attached to a fixed point O on a smooth horizontal table. The spring has natural
length 2 m and modulus of elasticity 21.6 N. The particle P is placed on the table at the point
A, where OA = 2 m. The particle P is now pulled away from O to the point B, where OAB is a
straight line with OB = 3.5 m. It is then released from rest.
(a)
Prove that P moves with simple harmonic motion of period
(b)
Find the speed of P when it reaches A.
π
3
s.
The point C is the mid-point of AB.
(c)
Find, in terms of π, the time taken for P to reach C for the first time.
Later in the motion, P collides with a particle Q of mass 0.2 kg which is at rest at A.
After the impact, P and Q coalesce to form a single particle R.
(d)
Show that R also moves with simple harmonic motion and find the amplitude of this
motion.
43. A car of mass 800 kg moves along a horizontal straight road. At time t seconds, the resultant
48000
newtons, acting in the direction of the motion
force acting on the car has magnitude
(t + 2) 2
of the car. When t = 0, the car is at rest.
(a)
Show that the speed of the car approaches a limiting value as t increases and find this
value.
(b)
Find the distance moved by the car in the first 6 seconds of its motion.
25
Level N Mechanics Top 60 questions
44. Figure
A uniform lamina occupies the region R bounded by the x-axis and the curve
y = sin x, 0 ≤ x ≤ π
as shown in Figure 2.
(a)
Show, by integration, that the y-coordinate of the centre of mass of the lamina is
π
8
.
Figure 3
A uniform prism S has cross-section R. The prism is placed with its rectangular face on a table
which is inclined at an angle θ ° to the horizontal. The cross-section R lies in a vertical plane
as shown in Figure 3. The table is sufficiently rough to prevent S sliding. Given that S does
not topple,
(b)
find the largest possible value of θ.
26
Level N Mechanics Top 60 questions
45. A light spring of natural length L has one end attached to a fixed point A. A particle P of mass
m is attached to the other end of the spring. The particle is moving vertically. As it passes
through the point B below A, where AB = L, its speed is ( 2 gL ) . The particle comes to
instantaneous rest at a point C, 4L below A.
(a)
Show that the modulus of elasticity of the spring is
8mg
.
9
At the point D the tension in the spring is mg.
(b)
Show that P performs simple harmonic motion with centre D.
(c)
Find, in terms of L and g,
(i)
the period of the simple harmonic motion,
(ii)
the maximum speed of P.
46. Figure
In a game at a fair, a small target C moves horizontally with simple harmonic motion between
the points A and B, where AB = 4L. The target moves inside a box and takes 3 s to travel from
A to B. A player has to shoot at C, but C is only visible to the player when it passes a window
PQ, where PQ = b. The window is initially placed with Q at the point as shown in Figure 4.
The target C takes 0.75 s to pass from Q to P.
(
)
(a)
Show that b = 2 − 2 L .
(b)
Find the speed of C as it passes P.
27
Level N Mechanics Top 60 questions
Figure 5
For advanced players, the window PQ is moved to the centre of AB so that AP = QB, as
shown in Figure 5.
(c)
Find the time, in seconds to 2 decimal places, taken for C to pass from Q to P in this
new position.
47.
Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope ABCD. The path
lies in a vertical plane. From A to B, the path is modelled as a straight line inclined at 60° to
the horizontal. From B to D, the path is modelled as an arc of a vertical circle of radius 50 m.
The lowest point of the arc BD is C.
At B, the skier is moving downwards with speed 20 m s−1. At D, the path is inclined at 30° to
the horizontal and the skier is moving upwards. By modelling the slope as smooth and the
skier as a particle, find :
(a)
the speed of the skier at C
(b)
the normal reaction of the slope on the skier at C
(c)
the speed of the skier at D
(d)
the change in the normal reaction of the slope on the skier as she passes B.
The model is refined to allow for the influence of friction on the motion of the skier.
(e)
State briefly, with a reason, how the answer to part (b) would be affected by using
such a model. (No further calculations are expected.)
28
Level N Mechanics Top 60 questions
48. A particle P of mass 0.8 kg is attached to one end A of a light elastic spring OA, of natural
length 60 cm and modulus of elasticity 12 N. The spring is placed on a smooth horizontal
table and the end O is fixed. The particle P is pulled away from O to a point B, where
OB = 85 cm, and is released from rest.
2π
s.
5
(b)
Find the greatest magnitude of the acceleration of P during the motion.
Two seconds after being released from rest, P passes through the point C.
(c)
Find, to 2 significant figures, the speed of P as it passes through C.
(d)
State the direction in which P is moving 2 s after being released.
(a)
49.
Prove that the motion of P is simple harmonic with period
Figure
A
2a
30°
O
P
A particle P of mass m is attached to one end of a light inextensible string of length 2a. The
other end of the string is fixed to a point A which is vertically above the point O on a smooth
horizontal table. The particle P remains in contact with the surface of the table and moves in a
⎛ kg ⎞
⎜ ⎟ , where k is a constant. Throughout the
⎝ 3a ⎠
motion the string remains taut and ∠APO = 30°, as shown in Figure 3.
circle with centre O and with angular speed
(a)
(b)
(c)
2kmg
.
3
Find, in terms of m, g and k, the normal reaction between P and the table.
Deduce the range of possible values of k.
Show that the tension in the string is
⎛ 2g ⎞
⎜
⎟ . The particle P now moves in a horizontal circle
⎝ a ⎠
above the table. The centre of this circle is X.
(d)
Show that X is the mid-point of OA.
The angular speed of P is changed to
29
Level N Mechanics Top 60 questions
50. A toy car of mass 0.2 kg is travelling in a straight line on a horizontal floor. The car is
modelled as a particle. At time t = 0 the car passes through a fixed point O. After t seconds the
speed of the car is v m s−1 and the car is at a point P with OP = x metres. The resultant force
1
on the car is modelled as
x(4 – 3x) N in the direction OP. The car comes to instantaneous
10
rest when x = 6. Find
(a)
an expression for v2 in terms of x,
(b)
the initial speed of the car.
51.
Figure
H
h
C
r
O
A body consists of a uniform solid circular cylinder C, together with a uniform solid
hemisphere H which is attached to C. The plane face of H coincides with the upper plane face
of C, as shown in Figure 2. The cylinder C has base radius r, height h and mass 3M. The mass
of H is 2M. The point O is the centre of the base of C.
(a)
Show that the distance of the centre of mass of the body from O is
14h + 3r
.
20
The body is placed with its plane face on a rough plane which is inclined at an angle α to the
horizontal, where tan α = 43 . The plane is sufficiently rough to prevent slipping. Given that the
body is on the point of toppling,
(b)
find h in terms of r.
30
Level N Mechanics Top 60 questions
52. Figure
O
A trapeze artiste of mass 60 kg is attached to the end A of a light inextensible rope OA of
length 5 m. The artiste must swing in an arc of a vertical circle, centre O, from a platform P to
another platform Q, where PQ is horizontal. The other end of the rope is attached to the fixed
point O which lies in the vertical plane containing PQ, with ∠POQ = 120° and
OP = OQ = 5 m, as shown in Figure 6.
As part of her act, the artiste projects herself from P with speed 15 m s–1 in a direction
perpendicular to the rope OA and in the plane POQ. She moves in a circular arc towards Q. At
the lowest point of her path she catches a ball of mass m kg which is travelling towards her
with speed 3 m s–1 and parallel to QP. After catching the ball, she comes to rest at the point Q.
By modelling the artiste and the ball as particles and ignoring her air resistance, find
(a)
the speed of the artiste immediately before she catches the ball,
(b)
the value of m,
(c)
the tension in the rope immediately after she catches the ball.
53. A rocket is fired vertically upwards with speed U from a point on the Earth’s surface. The
rocket is modelled as a particle P of constant mass m, and the Earth as a fixed sphere of
radius R. At a distance x from the centre of the Earth, the speed of P is v. The only force
cm
acting on P is directed towards the centre of the Earth and has magnitude 2 , where c is a
x
constant.
⎛1 1 ⎞
Show that v2 = U 2 + 2c ⎜ − ⎟ .
⎝ x R⎠
The kinetic energy of P at x = 2R is half of its kinetic energy at x = R.
(a)
(b)
Find c in terms of U and R.
31
Level N Mechanics Top 60 questions
54. Figure
y
x
0
1
(x – 2)2, the x-axis
2
and the y-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid S
is made by rotating R through 360° about the x-axis. Using integration,
The shaded region R is bounded by part of the curve with equation y =
(a)
calculate the volume of the solid S, leaving your answer in terms of π,
(b)
show that the centre of mass of S is
1
cm from its plane face.
3
Figure 4
A tool is modeled as having two components, a solid uniform cylinder C and the solid S. The
diameter of C is 4 cm and the length of C is 8 cm. One end of C coincides with the plane face
of S. The components are made of different materials. The weight of C is 10 W newtons and
the weight of S is 2 W newtons. The tool lies in equilibrium with its axis of symmetry
horizontal on two smooth supports A and B, which are at the ends of the cylinder, as shown in
Fig. 4.
(c)
Find the magnitude of the force of the support A on the tool.
32
Level N Mechanics Top 60 questions
55.
A particle P of mass m is attached to one end of a light inextensible string of length a. The
other end of the string is fixed at a point O. The particle is held with the string taut and OP
3
horizontal. It is then projected vertically downwards with speed u, where u2 = ga . When OP
2
has turned through an angle θ and the string is still taut, the speed of P is v and the tension in
the string is T, as shown in Fig.
(a)
Find an expression for v2 in terms of a, g and θ.
(b)
Find an expression for T in terms of m, g and θ.
(c)
Prove that the string becomes slack when θ = 210°.
(d)
State, with a reason, whether P would complete a vertical circle if the string were
replaced by a light rod.
After the string becomes slack, P moves freely under gravity and is at the same level as O
when it is at the point A.
(e)
Explain briefly why the speed of P at A is
⎛3 ⎞
⎜ ga ⎟ .
⎝2 ⎠
The direction of motion of P at A makes an angle φ with the horizontal.
(f)
Find φ .
33
Level N Mechanics Top 60 questions
56.
Figure
A
60°
P
FN
A particle P of mass 0.8 kg is attached to one end of a light inelastic string, of natural length
1.2 m and modulus of elasticity 24 N. The other end of the string is attached to a fixed
point A. A horizontal force of magnitude F newtons is applied to P. The particle P in in
equilibrium with the string making an angle 60° with the downward vertical, as shown in
Figure .
Calculate
(a)
the value of F,
(b)
the extension of the string,
(c)
the elasticity stored in the string.
57. Above the earth’s surface, the magnitude of the force on a particle due to the earth’s gravity is
inversely proportional to the square of the distance of the particle from the centre of the earth.
Assuming that the earth is a sphere of radius R, and taking g as the acceleration due to gravity
at the surface of the earth,
prove that the magnitude of the gravitational force on a particle of mass m when it is a
mgR 2
distance x (x ≤ R) from the centre of the earth is
.
x2
(a)
A particle is fired vertically upwards from the surface of the earth with initial speed u, where
3
u2 = gR . Ignoring air resistance,
2
(b)
find, in terms of g and R, the speed of the particle when it is at a height 2R above the
surface of the earth.
34
Level N Mechanics Top 60 questions
58. A car moves round a bend which is banked at a constant angle of 10° to the horizontal. When
the car is travelling at a constant speed of 18 m s−1, there is no sideways frictional force on the
car. The car is modelled as a particle moving in a horizontal circle of radius r metres.
Calculate the value of r.
59. A particle P of mass 2.5 kg moves along the positive x-axis. It moves away from a fixed
origin O, under the action of a force directed away from O. When OP = x metres the
magnitude of the force is 2e−0.1x newtons and the speed of P is v m s−1. When x = 0, v = 2. Find
(a)
v2 in terms of x,
(b)
the value of x when v = 4.
(c)
Give a reason why the speed of P does not exceed
20 m s−1.
60. Figure
A
A light inextensible string of length 8l has its ends fixed to two points A and B, where A is
vertically above B. A small smooth ring of mass m is threaded on the string. The ring is
moving with constant speed in a horizontal circle with centre B and radius 3l, as shown in Fig.
2. Find
(a)
the tension in the string,
(b)
the speed of the ring.
(c)
State briefly in what way your solution might no longer be valid if the ring were firmly
attached to the string.
35