Download Problems - Bartholomew Andrews

Mechanics Problem Sheet 1: Kinematics
1. A particle’s displacement is given by
m. Derive expressions for the
particle’s velocity, v, and acceleration, a, and plot graphs of a, v and x against time from 0 to
5 s. Compute the particle’s average velocity between 2 and 3 s, and compare this with its
instantaneous velocity at 2.5 s. Comment on your answer. (-3 ms-1, -3 ms-1)
2. The position of a particle is given by
. Derive expressions for the particle’s
velocity, v, and acceleration, a, and plot graphs of a, v and x against time. Compute the
particle’s average velocity between 1 and 3 s, and compare this with its instantaneous
velocity at 2 s. What happens to a, v and x as the time becomes very large? Describe the
particle’s motion in words. (-1.35 ms-1, -1.34 ms-1)
3. A bus starts from rest at the origin and accelerates at 2 ms -2 for 3 s. It moves at constant
velocity for 2 s and then has an acceleration of -3 ms-2 for 2 s. Plot graphs of a, v and s
against time. What is the total distance covered? (27 m)
4. A cyclist is initially moving at 12 ms-1. He covers 32 m in the next 4 s. Assuming a constant
acceleration find (a) the acceleration and (b) his speed after 4 s. (-2 ms-1, 4 m)
5. A car has initial velocity of 20 ms-1 and an acceleration of -5 ms-2. Find its average velocity
in the time interval during which its displacement is 30 m from the initial position. (15 ms-1)
6. A tennis ball is dropped from rest from a height of 5.00 m and rebounds to a height of 3.20
m. If it is in contact with the floor for 0.0360 s, what is its average acceleration during this
period? Without detailed calculation, describe what form the actual acceleration is likely to
take. Take g = 9.81 ms-2. (495 ms-2 upwards)
7. An object is fired vertically up from the ground. Find its maximum height and the time of
flight given that it loses 30.0 % of its initial speed in 1.80 s and is moving upward. Take g =
9.81 ms-2. (177 m, 12.0 s)
Vijay Tymms, Mechanics PS1: Kinematics, October 2010
8. A particle is projected vertically upwards from the Earth’s surface with speed u and at the
same instant, a second particle is released from rest at a point vertically above the first
particle and a height h above the Earth’s surface. Taking g to be a constant show that the
particles will collide when at a height of
. Find the condition such that the first
particle is instantaneously at rest when the particles collide.
9. A particle is constrained to move in a straight line slows with an acceleration which is
proportional to the square of the speed of the particle. Show that the time required to slow
the particle to one-quarter of its original speed is three times that required to slow it to half
its original speed, and prove that the distances travelled are in the ratio 2:1
Vijay Tymms, Mechanics PS1: Kinematics, October 2010
Mechanics Problem Sheet 2: Statics and Dynamics
Use g = 9.81 ms-2 throughout this problem sheet
1. A 2.00 kg block is suspended by a single rope. A horizontal force holds the rope at 37.0⁰ to the
vertical. Draw a free body diagram for the block and find a) the force and b) the tension in the rope.
(14.8 N, 24.6 N)
2. Two blocks with masses mA = 0.200 kg and mB = 0.300 kg hang under one another as shown
below:
Find the tension in the (massless) ropes in the following situations: a) The blocks are at rest; b) they
move upwards at 5.00 ms-1; c) they accelerate upwards at 2.00 ms-2; d) they accelerate downwards at
2 ms-2. e) If the maximum allowable tension in the rope is 10.0 N, what is the maximum possible
upward acceleration? (a) 4.9 N, 2.94 N, b) 4.9N, 2.94 N, c) 5.9 N, 3.54 N, d) 3.9 N, 2.34 N, e)
10.2 ms-2)
3. A person of mass 75.0 kg stands on a set of scales in an elevator. The scales record the normal
contact force between the person and the floor. What can you infer about the motion of the
elevator if the scales read a) 736 N b) 600 N c) 900 N d) 0 N? These readings are referred to as the
person’s apparent weight. For situation (d) the person is said to be “weightless”. Comment on this
terminology. (Constant v, 1.8 ms-2 down, 2.2 ms-2 up, g down)
4. A child pulls a 3.60 kg sled at 25.0⁰ to a slope that is at 15.0⁰ to the horizontal. The sled moves at
constant velocity when the tension is 16.0 N. a) Draw a free body diagram for the sled and compute
the coefficient of friction between the sled and the ground. b) What is the acceleration of the sled if
the rope is released? ((a) 0.196, (b) 0.682 ms-2)
Vijay Tymms, Mechanics PS2: Particle Dynamics, October 2010
5. (Adapted from Michael Coppins’ lecture notes): A horse is required to pull a cart. Unfortunately,
not only is the horse rather lazy, but he believes he has some understanding of Classical Physics. He
refuses to cooperate, arguing that by Newton’s third law the force he exerts on the cart will be
exactly balanced by the equal and opposite force that the cart exerts on him. Thus any attempt to
pull the cart would be futile. How would you explain the fallacy of this argument to the horse. NB
the horse is particularly receptive to free body diagrams (and sugar lumps, though they have little to
do with the discussion...)
6. A 60.0 kg parachutist and her 7.00 kg parachute fall at a constant 6.00 ms-1. Find (a) the force on the
woman due to the chute; (b) the force on the chute due to the air. (Ignore the force on the woman
due to the air.) (588 N upwards, 657 N upwards)
7. Two particles, each of mass m, are attached to ends of a light string. The string passes over two
smooth pegs in a vertical wall, the line joining the pegs making an angle θ with the horizontal. Show
that the contact force by the string on the upper peg and the contact force by the string on the
lower peg are perpendicular to each other.
8. A frictionless chain of length L slides off a frictionless table. Take the y coordinate as y = 0 at the
table’s surface and downwards as positive. Initially a length y0 hangs over the edge. a) Find the
acceleration of the chain as a function of y. b) Show that the velocity of the chain as a function of
the lowest point of the chain, y, is
.
c) Optional. Hence show that the vertical displacement of the base of the chain is given by
and verify that the chain barely moves until t approaches
and then rushes off the table.
NB the physics of part c) is no trickier than b) but the mathematics is more complicated. It requires knowledge of the
hyperbolic cosine function which some of you will not yet have met, and uses the identity
Vijay Tymms, Mechanics PS2: Particle Dynamics, October 2010
Mechanics Problem Sheet 3: Mechanical Energy and Gravitation
1. A 500 g block is dropped from a height of 60 cm above the top of a vertical spring whose
stiffness is k = 120 Nm-1. Find the maximum compression. Assume the block sticks to the
spring when it hits. (26.6 cm)
2. A ski lift carries people at 1 ms-1 for a distance of 0.5 km up a 200 slope. The chairs are 5 m
apart and each carries a single skier. If the chairs are all occupied, what power is required?
Assume that the average mass of a skier if 70 kg. (23.4 kW)
3. Find the potential energy function U(x) corresponding to the force
.(
. Take
)
4. An object is fired vertically from the Earth with speed
where vesc is the Earth’s escape
velocity and N > 1. Show that the maximum altitude is
where RE is the Earth’s radius. Ignore the Earth’s rotation and air resistance.
5. The period T of a simple pendulum of length L is given by
where g is the field
strength. It is adjusted to keep the correct time at sea level where g = 9.810 Nkg-1. When it
is taken to a mountain top it loses 1 minute per day. a) What is g at this point? b) What is
the height of the mountain? Take the Earth’s radius as 6.37 x 106 m (9.796 ms-2, 4.42 km)
6. A composite object consisting of two spheres (each of radius a and mass m) joined together,
falls under gravity towards a supermassive black hole of mass M. It falls so that the axis
joining the two centre of the spheres points towards the black hole. The centre of the
nearer sphere is distance r from the black hole.
a) Assuming that
and using a binomial expansion show that the acceleration due to
gravity caused by the black hole at the position of the other (further) sphere is
approximately
.
Vijay Tymms, Mechanics PS3: Energy and Gravity, November 2010
b) The fact that the nearer sphere experiences a stronger gravitational force means there is
an effective force trying to break the bond joining the spheres. Such forces are called
tidal forces. Show that
(i.e. if the force holding the spheres together is
less than Ftidal then they will be torn apart).
7. The potential energy shared by two atoms separated by a distance r in a diatomic molecule
is given by the Lennard-Jones function:
where U0 and r0 constants. Sketch U(r), identifying where
occurs, prove that the
minimum potential is –U0 at r0 (what is the physical significance of this point?) and
determine where the corresponding force is zero. (r = r0/21/6, r = r0)
8. An Eskimo child slides on an icy (frictionless) hemispherical igloo of radius R starting with
negligible speed at the top.
a) At what angle with the vertical does she lose contact with the surface? (cos-1(2/3))
b) If there were friction would contact be lost at a higher or lower point?
9. Consider two identical bodies of mass M, fixed at
. Find an expression for the
potential energy of a third identical mass free to move along the x axis. Sketch the potential
function as a function of x, identify where the singularities are and show there is an
equilibrium point at
. Verify that this equilibrium point is unstable.
Vijay Tymms, Mechanics PS3: Energy and Gravity, November 2010
Mechanics Problem Sheet 4: Noninertial Reference Systems
Use g = 9.81 ms-2 throughout
Linear Acceleration
1. A 75 kg person stands on a bathroom scale while riding in an elevator. If the elevator has
(a) upward and (b) downward acceleration of g/4 what is the weight indicated on the scale
in each case? (920 N, 552 N)
2. An ultracentrifuge has a rotational speed of 500 rps. (a) Find the centrifugal force on a 1 μg
particle in the sample chamber if the particle is 5 cm from the rotation axis. (b) Express the
result as a ratio of the centrifugal force to the weight of the particle. Explain why the term
“centrifugal force” is a misnomer. (493 μN away from the centre, 50,300)
3. A plumb line is held steady while being carried along in a moving train. If the mass of the
plumb bob is m = 2kg, find the tension in the cord and the deflection from the local vertical
if the train accelerates forward at g/10. (19.7 N, about 5.7⁰)
4. If in problem 3, the plumb line is not held steady but oscillates as a simple pendulum, find
the period of (small) oscillations. The length of the pendulum is 20 cm. (0.895 s)
5. A truck is travelling on a level road. The driver applies the brakes causing the truck to
decelerate by g/2. This causes a box in the rear to slide forward. If the coefficient of sliding
friction between the box and the truck floor is 1/3 find the acceleration of the box relative
to (a) the truck and (b) the road. (g/6 forward, g/3 backward)
Vijay Tymms, Mechanics PS4: Noninertial Reference Systems, November 2010
Rotating Systems
6. A batsman at Lord’s (latitude 51⁰ N) drives a ball a distance of 50 m in a fairly flat
trajectory. A fielder tries to catch the ball but misses it and claims the ball was deflected due
to the Coriolis force hence the butterfingers... Is his argument valid?
7. A racer drives at 400 mph due North on salt flats at Utah (latitude 41⁰ N). Find the ratio of
the magnitude of the Coriolis force on the car to its weight. What is the Coriolis force’s
direction? (9.6 x 10-4, deflected to East)
8. A bug crawls at constant speed on a circular path of radius b on a turntable rotating at
constant angular speed ω. The circular path is concentric with the turntable. If the mass of
the insect is m and the coefficient of static friction between bug and turntable is μ, how fast,
relative to the turntable can the bug travel if it goes in the direction of rotation of the
turntable? ( v = (μg – ω2b)/(2ω) )
Vijay Tymms, Mechanics PS4: Noninertial Reference Systems, November 2010
PS4: Corrections and Issues
Question 6
This question is very similar to that given in Classwork 3, question 2 on the snooker ball. There
is a mistake in the answers though – the deflection should be multiplied by the sine of the
latitude, not the cosine. This gives a deflection of 0.14128/velocity, which doesn’t really change
the outcome much. Note that an estimate of the velocity is deliberately not given.
Question 7
Two mistakes made on the answers. Again, sine of latitude, not cosine of latitude should be
used. Also, 400 mph is 177.78 ms-1. The final answer should be a ratio of 1.72 x 10-3.
Question 8
Answer may be wrong. A student has put a convincing argument forward to suggest that the
angular velocities should add vectorially to give a total of the angular velocity of the plate plus
v/b which gives a different result. Will come back to it...
Vijay Tymms, November 2010
Vijay Tymms, PS4 Issues, November 2010
Mechanics Problem Sheet 5: Rotational Statics and Dynamics
Use g = 9.81 ms-2 throughout
1. Support a uniform ruler with two fingers at different distances from the centre. Slowly
reduce the separation between the two fingers (as demonstrated in lectures). Explain what
happens. (NB in an exam this question would carry 5 or 6 marks i.e. there is plenty of physics to discuss.
A diagram is essential.)
2. Find the centre of mass of:
a) a system of four particles in the xy plane: m1 = 2 kg at (3,-1), m2 = 4 kg at (3,3), m3 = 5
kg at (-4,4) and m4 = 1 kg at (-3,-2) (-5/12,7/3)
b) A thin rod of length 3L bent at right angles a distance L from the end with respect to
the corner (L/6,2L/3)
c) A semicircular rod of radius R and linear density λ kgm-1. Hint: set up a coordinate
system so that the rod is symmetric about the y axis so the x centre of mass is at zero.
Then consider a small element of the rod and go on to integrate. (2R/π from the
“centre” of the semicircle)
3. A 75 kg man sits at the rear end of a platform of mass 25 kg and length 4 m moving at 4
ms-1 East over a frictionless surface. At t = 0 , he walks at 2 ms-1 relative to the platform and
then stops at the front end. Find the displacements, relative to the frictionless surface, of (i)
the platform, (ii) the man and (iii) the centre of mass of the system. (5m, 9m, 8m)
4. Two identical ladders, AB and BC, each of weight w are hinged together at B. Ends A and C
rest on a smooth, horizontal floor and are connected by a light, inextensible string so that
A, B and C form an equilateral triangle in a vertical plane. When a man whose weight is
four times that of either ladder stands halfway up ladder AB what are the contact forces
with the floor of A and C? Find the tension in the string and the contact force of ladder AB
on ladder BC. Hint: the tension and the AB contact forces are internal forces – they cannot
be resolved by considering the whole system. Instead, think about dividing the system into
two parts vertically and analyse that. (4w, 2w, (√3/2)w, (√7/2)w)
Vijay Tymms, Mechanics PS5: Rotation, November 2010
5. Find the moment of inertia of:
a) a uniform disc rotating in the plane about its centre, mass M of radius R with a hole
drilled into it of radius a where the centre of the hole is a distance b from the centre of
the disc. (M(R4 – a4 – 2a2b2)/(2(R2-a2))
b) a hollow cone of mass M, height h and apex angle 2θ about the central axis of symmetry
(1/2Mh2sec2θ)
6. Two particles, masses m1 and m2 are fixed to opposite ends of a light rod of length 2a.
Initially the rod is vertical and perpendicular to the surface of a smooth table on which
particle m1 rests. Describe in words what happens to the system if it is displaced slightly.
Find the angular velocity of the rod when it becomes horizontal. (√(g/a))
7. A meter stick of mass 40 g is pivoted at the 35 cm mark. What is its moment of inertia
about the pivot point and angular acceleration as a function of angle, θ, to the horizontal?
(4.23 gm2, 14cosθ rads-2)
Vijay Tymms, Mechanics PS5: Rotation, November 2010
PS5: Corrections and Issues
Question 3
Many students had difficulties with this question. The key thing to remember that as there are
no external force on the system, the velocity of the centre of mass of the system will remain
constant (at 4 ms-1). The third part of the question then becomes the quickest to answer (and
should therefore really be the first part asked on the question): as the man travels for 2 seconds
to get from one end of the platform to the other, the centre of mass of the system travels 8
metres. The other parts of the question can be tackled by working out the problem for a
stationary centre of mass and adding +8 m to the answers.
Question 5
For part (a) remember that the mass M is the mass of the disc with the hole in it, not the mass
of the disc if it where whole
Question 6
The answer I have written is fine but perhaps needs spelling out a bit. When the rod topples the
centre of mass remains at its horizontal coordinate throughout the motion (there was a lecture
demonstration to illustrate this). The mass m1 slides outwards, starting at zero velocity, getting
faster, then slowing down to finish with zero velocity at the instant m2 collides with the table.
This means the rod has zero horizontal velocity (which is necessary as there are never any
horizontal forces on the rod). As the mass m1starts and ends with no kinetic energy, nor
gravitational potential energy (relative to the table top) the conservation of energy can applied
solely to mass m2. When this mass collides with the table as the bottom mass is stationary is can
be considered to be rotating about the base and its rotational KE must equal its GPE when
vertical. The mathematics is detailed on the answer sheet.
Vijay Tymms, December 2010
Vijay Tymms, PS5 Issues, December 2010
Mechanics Problem Sheet 6: Torque and Oscillations
1. A car with tires of radius 25 cm moves at constant velocity of 30 ms-1. Find the velocity
relative to the road of a) the top b) the base c) the forward edge and d) the backward edge
of the tires. (60 ms-1, 0, 42.4 ms-1 45° below vertical, 42.4 ms-1 above vertical)
2. A flywheel initially rotating at 1,200 rpm stops in 4 minutes when only friction acts. If an
additional torque of 300 Nm is applied it stops in 1 minute. a) What is the moment of
inertia of the flywheel? b) What is the frictional torque? ((1/6π) ≈ 53.1 gm2, 100 Nm)
3. A solid cylinder of mass M and radius R unwinds on a vertical string:
a) Use the Conservation of Energy to show that the speed of the spool after it falls a
distance h is
. b) Use the result from (a) to find the linear acceleration of the centre
of mass. c) Use dynamics to find the linear acceleration. d) What is the tension? e) With
what force should the string be pulled to have the spool spin but not fall? What is its
angular acceleration in this case? (2g/R rads-2)
4. A lawn roller is a solid cylinder of mass M and radius R. It is pulled at its centre by a
horizontal force P and rolls without slipping on a horizontal surface:
Find (a) the acceleration of the cylinder; (b) the frictional force acting on it. (2P/3m, P/3)
Vijay Tymms, Mechanics PS6: Rolling and Oscillations, November 2010
5. A coin rests on the top of a piston that executes simple harmonic motion vertically with an
amplitude of 10 cm. At what minimum frequency will the does the coin lose contact with piston
a) on Earth (gE = 9.81 Nkg-1) b) on the Moon (gM = gE/6)? (1.58 Hz, 0.644 Hz)
6. A block of mass m is placed on top of another block of mass M that is attached to a
horizontal spring of stiffness k. The coefficient of friction between the blocks is μ and the lower
block slides on a frictionless horizontal surface. The amplitude of oscillation is A:
Find an expression for the coefficient of friction required such that the upper block does not slip
relative to the lower block. ( kx/((M+m)g) )
7. A block of mass M is attached to a horizontal spring of mass m.
a) Show that when the speed of the block is v, the kinetic energy of the spring is
. Hint:
first consider the KE of an element of the length dx. What will its speed be? b) Use the
Conservation of Energy or otherwise to show that the period of oscillation is
Vijay Tymms, Mechanics PS6: Rolling and Oscillations, November 2010
.
PS6: Corrections and Issues
Question 2
Error on answers where I’ve converted rpm to radians by multiplying by 60 instead of dividing.
Corrected solution is now on the question sheet...
Vijay Tymms, December 2010
Vijay Tymms, PS6 Issues, December 2010
Mechanics Problem Sheet 7: Projectiles and Orbits
Notes
o For questions 1 – 4 on this problem sheet use refer to the symbols given on Section
3.5 Classical Orbits
o For questions 2 – 4 there is no need to start from first principles; you may use the
results proved in question 1
o Take g = 10 ms-2 unless stated otherwise
1. Starting from first principles, and stating any assumptions you make, prove the following
relations from projectile motion given in lectures:
a) Equation of motion:
b) Maximum height:
c) Range:
d) Time of flight:
What do the negative roots in the range and time of flight correspond to?
2. Two balls are thrown from the top of a cliff with equal initial speeds. One starts at angle θ
above the horizontal, while the other starts at θ below (i.e. –θ). Show that the difference in
their ranges is
. (The identity
may be of use.) Verify that
this is the case by trying sample values in the Parabola spreadsheet calculator.
3. a) Show that if the height, h, is set at zero then the maximum range will be when
and find the corresponding maximum range.
b) If the height is non-zero it can be proved that the angle for maximum range is
and the corresponding maximum range is
.
(You can prove these if you wish but it involves a lot of algebra and does not really provide
any extra physical insight so this remains an optional exercise.) Instead: (i) For a velocity of
50 ms-1 find the height for which the optimum range is at an angle of 60⁰ and find the
range, (ii) for a velocity of 50 ms-1 find the height for which the optimum range is at an
Vijay Tymms, Mechanics PS7: Projectiles and Orbits, December 2010
angle of 30⁰ and find the range and (iii) find approximations for optimal angle and range for
when
. In all cases comment on any interesting results and verify your findings
using the Parabola spreadsheet calculator. (250 m, -83.3 m, v.(2gh)-1/2, v(2h/g)1/2)
4. For a given velocity of projectile, provided it is sufficient to reach a certain specified range
over flat ground (i.e.
), there are, in general two possible angles of projection, θ1 and
θ2 that can reach this range where
. Find expressions for these two angles and show
that the ratio of the flights times is given by
and the ratio of the maximum
. Once again, use the Parabola spreadsheet calculator for
heights is
verification.
5. In its elliptic orbit, the speed of the Earth at perihelion is vP = 30.3 kms-1. If the distances to
the Sun at perihelion and aphelion are rP = 1.47 x 1011 m and rA = 1.52 x 1011 m, find vA.
The mass of the Sun is 1.99 x 1030 kg, G = 6.67 x 10-11 Nm2kg-2 (29.3 kms-1)
6. Many satellites move in a circle above the Earth’s equatorial plane. They are at such a height
above the surface that they always remain above the same point. (a) Find the altitude of
such “geosynchronous” satellites. (b) Explain with a sketch why the radio signals from
these satellites cannot directly reach receivers on Earth above a latitude of 81.3⁰ North. The
Earth’s mass is 5.99 x 1024 kg, radius is 6.0 x 106 km. (36,300 km)
7. Three stars of equal mass m rotate in a circular path of radius r about their centre of mass:
They are equidistant from each other. Show that the angular velocity of the motion is given
by
.
Vijay Tymms, Mechanics PS7: Projectiles and Orbits, December 2010
Mechanics Problem Sheet 8: Solids and Fluids
Density
1. The antifreeze in a radiator consists of 70 % ethylene glycol of density 0.8 gcm -3 and 30 %
water. Find the density of the mixture if the percentages refer to (a) volume and (b) mass.
(Ignore the fact that the volume of the mixture is somewhat less than the sum of the origina
volumes.) (0.860 gcm-3, 0.851 gcm-3)
Elastic Moduli
2. A circular steel wire of length 1.8 m must not stretch by more than 1.5 mm when a load of
400 N is applied. What is the minimum diameter required? The Young’s modulus of steel is
200 GPa; ignore the weight of the wire. (1.75 mm)
3. The pressure at the bottom of the Marianas Trench in the Pacific Ocean is about 108 MPa.
What is the fractional change in volume if a given mass of water is moved from the surface
to this depth? The bulk modulus of sea water is 2.1 GPa. (5.14 %)
Pressure in Fluids
4. The piston in a hypodermic syringe has a radius of 0.5 cm, and the needle has a hole of
radius 0.15 mm. What force must be applied to the plunger in order to inject fluid into a
vein in which the blood pressure is 20 mm Hg (0.21 N)
5. The gauge pressure in the tyres of a car is 200 kPa. The area of each tire in contact with the
road is 120 cm2. What is the mass of the car? (980 kg)
Archimedes Principle
6. A glass test tube of radius 0.8 cm is weighted at the base and this floats vertically in water.
What additional mass dropped into the tube would cause it to sink a further 3 cm? (6.03 g)
7. A sphere floats in water with 60 % of its volume submerged. It floats in oil with 70 % of its
volume submerged. What is the density of the oil? (857 kgm-3)
Vijay Tymms, Mechanics PS8: Solids and Fluids, December 2010
Equation of Continuity
8. Water flows at 1.2 ms-1 through a hose of diameter 1.59 cm. How long does it take to fill a
cylindrical pool of radius 2 m to a height of 1.25 m? (1100 minutes)
Bernoulli Equation
9. Water enters a basement inlet pipe of radius 1.5 cm at 40 cms-1. It flows through a pipe of
radius 0.5 cm at a height of 35 m at a gauge pressure of 0.2 atm. (a) What is the speed of the
water at the higher point? (b) What is the gauge pressure at the basement? (3.6 ms-1, 370
kPa)
Vijay Tymms, Mechanics PS8: Solids and Fluids, December 2010
Mechanics Problem Sheet 9: Pulleys, Particle Interaction and Rockets
Pulleys and Tension
1. In an equal-arm arrangement, a mass 5m is balanced by the masses 3m and 2m, which are
connected by a string over a pulley of negligible mass and prevented from moving by a
string A:
Analyze what happens if the string A is suddenly severed (by e.g. a lighted match).
Elastic Collisions
and initial velocity
2. A block of mass
with a block of mass
moving at
makes a 1D elastic collision
. Find their final velocities and
confirm that kinetic energy is conserved in the collision. (1.6 ms-1, 3.6 ms-1)
3. A particle of mass M1 collides elastically with a particle of mass M2 ( > M1) initially at rest.
Show that the maximum angle θ1 to the original direction of motion at which M1 can move
off is given by
. (Hint: Obtain a quadratic equation in v1. What is the condition
for a real solution?)
Inelastic Collisions
4. A 90 kg rugby player runs due North at 8 ms-1 to tackle a 110 kg player running East at 7.5
ms-1. If the completely inelastic collision occurs while their feet are briefly off the ground,
find (a) their common velocity just after the collision and (b) the loss in kinetic energy.
(4.13i + 3.60 j ms-1, 2.97 kJ)
5. Two particles with masses m1 and m2 travel toward each other with speeds u1 and u2. They
collide and coalesce. Show that the loss in kinetic energy is
Vijay Tymms, Mechanics PS9: Pulleys and Collisions, December 2010
.
Simple Explosions
6. A 10 kg object with a velocity 6i ms-1 explodes into two equal fragments. One flies off with
a velocity 2i – j ms-1. What is the velocity of the other fragment? (10i + j ms-1)
7. The mass of a rocket without fuel is
and the exhaust speed of the gases is
relative to the rocket. If the rocket starts at rest with respect to the Earth,
what mass of fuel MF is required to reach the exhaust speed? (1.7 x 104 kg)
Vijay Tymms, Mechanics PS9: Pulleys and Collisions, December 2010