Download FEP Prep for Unit test

Force, Energy & Periodic
Motion
Preparation for unit test
Advanced Higher
Mathematics of Mechanics
Summary of assessment standards (Unit assessment standard only)
In the unit test you can expect to be asked at least one question on each sub-skill.
1.1 Applying skills to principles of momentum, impulse, work, power and energy
Sub-skills
Unit Assessment standard
Working with impulse as the
 Use impulse appropriately in a simple situation, making use of the equations
change in momentum, and/or
𝑰 = 𝑚𝒗 − 𝑚𝒖 = ∫ 𝑭 𝑑𝑡 and 𝐼 = 𝐹𝑡
force as the rate of change of
momentum
Working with the concept of
 Use the concept of the conservation of linear momentum:
conservation of linear
𝑚1 𝒖1 + 𝑚2 𝒖2 = 𝑚1 𝒗1 + 𝑚2 𝒗2 or 𝑚1 𝒖1 + 𝑚2 𝒖2 = (𝑚1 + 𝑚2 )𝒗
momentum
Determining work done by a
 Evaluate appropriately the work done by a constant force, making use of the
constant force in one or two
equation 𝑊 = 𝐹𝑠 (one dimension)
dimensions, or a variable force
 Apply to practical examples the concept of power as the rate of doing work:
𝑑𝑊
during rectilinear motion
𝑃=
=𝑭∙𝒗
𝑑𝑡
1
Using the concepts of kinetic
 𝐸𝐾 = 2 𝑚𝑣 2, 𝐸𝑃 = 𝑚𝑔ℎ for a uniform gravitational field.
(EK) and/or potential (EP) energy
Work done = change in energy
to applying the work-energy
principle
Using the concepts of kinetic
 𝐸𝐾 + 𝐸𝑃 is constant for simple problems involving motion in a plane
(EK) and/or potential
(EP) energy within the concept
of conservation of energy
1.2 Applying skills to motion in a horizontal circle with uniform angular velocity
Sub-skills
Unit Assessment standard
Applying equations to motion in  Solve problems involving motion in a circle of radius 𝑟, with uniform angular
a horizontal circle with uniform velocity 𝜔, making use of the equations:
angular velocity
𝜃 = 𝜔𝑡,
𝑣 = 𝑟𝜔 = 𝑟𝜃̇,
𝑣2
𝑎 = 𝑟𝜔2 = 𝑟𝜃 2̇ = ,
𝒂 = −𝜔2 𝒓,
2𝜋
𝑇= 𝜔
𝑟
𝐺𝑀𝑚
Using equations for horizontal
 Solve a simple problem using Newton's Inverse Square Law, 𝐹 = 2 , 𝑎 =
𝑟
circular motion alongside
𝑘
Newton's Inverse Square Law of 𝑑2
Gravitation
1.3 Applying skills to Simple Harmonic Motion
Sub-skills
Unit Assessment standard
Working with the concept of
 Understand the concept of SHM and use the basic equation 𝑥̈ = −𝜔2 𝑥, and
Simple Harmonic Motion (SHM) the following associated equations, knowing when and where they arise in
order to solve basic problems involving SHM in a straight line:
𝑣 2 = 𝜔2 (𝑎2 − 𝑥 2 ) where 𝑣 = 𝑥̇ ,
2𝜋
𝑇= 𝜔 ,
|𝑣|𝑚𝑎𝑥 = 𝜔𝑎,
|𝑥̈ |𝑚𝑎𝑥 = 𝜔2 𝑎
𝑥
Applying Hooke's Law to
 Make use of the equation for Hooke's Law 𝑇 = 𝑙 , to determine an unknown
problems involving SHM
tension/thrust, modulus of elasticity or extension/compression of natural
length.
Advanced Higher
Mathematics of Mechanics
1.4 Applying skills to Centres of Mass
Sub-skills
Unit Assessment standard
Determining the turning effect
 Evaluate the turning effect of a single force or a set of forces acting on a
of force
body, considering clockwise and anticlockwise rotation:
Moment of force about point P = magnitude of force × perpendicular distance from P
Using moments to find the
centre of mass of a body
Advanced Higher
and/or
Understand that for a body in equilibrium the sum of the moments of the
forces about any point is zero
 Equate the moments of several masses acting along a line to that of a single
mass acting at a point on the line
∑ 𝑚1 𝑥1 = 𝑥̅ ∑ 𝑚1 where (𝑥̅ , 0) is the centre of mass of the system
 Extend this to two perpendicular directions to find the centre of mass of a
set of particles arranged in a plane.
∑ 𝑚1 𝑥1 = 𝑥̅ ∑ 𝑚1 and ∑ 𝑚1 𝑦1 = 𝑦̅ ∑ 𝑚1 where (𝑥̅ , 𝑦̅) is the centre of mass of
the system
Mathematics of Mechanics
Force, Energy & Periodic Motion – Preparation for Assessment
Assessment Standard 1.1: Applying skills to principles of momentum, impulse, work, power and energy
1)
A force of (2𝒊 + 7𝒋) newtons acts on a body of mass 5 kg for 10 seconds.
The body was initially moving with a constant velocity of (𝒊 − 2𝒋) m s-1.
Find the final velocity of the body in vector form, and hence obtain its final speed.
2)
A bullet of mass 40 grams, travelling at 80 m s-1, hits a stationary block of wood of mass 2 kg which
is free to move on a smooth, horizontal plane.
Use the conservation of linear momentum to calculate the final speed of the block, given that the
bullet passes right through the block and emerges with speed 20 m s-1.
3)
A constant force acting on a particle of mass 3 kg increases its speed from 2 m s-1 to 10 m s-1 in
6 seconds, where the particle is free to move in the direction of the constant force.
Calculate the change in linear momentum of the particle and hence find the magnitude of the
constant force.
4)
A bullet of mass 50 grams, travelling at 90 m s-1, hits a stationary block of wood of mass 3 kg which
is free to move on a smooth, horizontal plane.
Given that the bullet becomes embedded in the block, use the conservation of linear momentum to
find the speed of the block and bullet immediately after the collision.
5)
A body of mass 200 g is moving with velocity (40𝒊 − 30𝒋) m s-1 when an impulse is applied.
The impulse causes the velocity to change to (10𝒊 + 15𝒋) m s-1.
Find the impulse.
6)
A bullet is fired from a gun with a horizontal velocity of 400 m s-1. The mass of the gun is 3 kg and
the mass of the bullet is 60 g.
Find the initial speed of the recoil of the gun and the gain in kinetic energy of the system.
State any assumptions made.
7)
Two identical railway trucks are travelling in the same direction along the same straight piece of
track with constant speeds of 6 m s-1 and 2 m s-1. The faster truck catches up with the other one
and on collision the two trucks automatically couple together.
Find the common speed of the trucks after collision.
8)
A body S of mass 3 kg is moving with velocity (−2𝒊 + 5𝒋) m s-1 when it collides with a body T,
of mass 5 kg moving with velocity (3𝒊 − 6𝒋) m s-1.
Immediately after the collision the velocity of S is (3𝒊 − 5𝒋) m s-1.
Find the velocity of T immediately after the collision.
Advanced Higher
Mathematics of Mechanics
9)
A body of mass m kilograms falls from rest at point P, down a smooth chute, to point Q,
as shown in the diagram. The vertical distance of P above Q is 10 m.
Calculate the speed of the body when it reaches Q.
State any assumptions you have made.
10)
A body of mass 6 kg is released from rest at a point P on a rough plane inclined at an angle sin-1(¼)
to the horizontal.
After travelling a distance of 10 m down the line of greatest slope of the plane to a point Q, the
body acquires a speed of 4 m s-1.
Calculate the work done against friction in travelling from P to Q.
11)
A car of mass 1500 kg moves along a straight, horizontal road against a constant resistance of
2000 newtons. Given that the power at which the car's engine is working is 16 kW, calculate the
acceleration of the car at the instant when its speed is 4 ms-1.
12)
A car of mass 1200 kg moves along a straight, horizontal road against a constant resistance of
1900 newtons. Given that the power at which the car's engine is working is 15 kW, calculate the
speed of the car at the instant when its acceleration is 0·5 ms-2.
13)
A car of mass 1600 kg moves along a straight, horizontal road against a constant resistance of
2400 newtons. Given that the car is accelerating at 0·8 m s-2, calculate the power at which the car's
engine is working at the instant when the car's speed is 5 m s-1.
14)
A body of mass 10 kg is released from rest at a point P on a rough plane inclined at an angle
1
sin−1 (3) to the horizontal. After travelling a distance of 12 m down the line of greatest slope of
the plane to a point Q, the body acquires a speed of 7 m s-1.
Calculate the work done against friction in travelling from P to Q.
15)
From a point situated at the bottom of a rough inclined plane, a body is projected with a speed of
4
56 m s-1 along and up a line of greatest slope. The plane is inclined at tan−1 (3) to the horizontal.
4
The coefficient of friction between the body and the plane is 7.
How far along the slope will the body travel before it comes to rest?
16)
A body of mass 2 kg falls vertically against a constant resistance of 14 newtons. The body passes
through two points A and B when travelling with speeds 3 m s-1 and 10 m s-1 respectively.
Find the distance AB.
17)
A rough slope of length 5m is inclined at an angle of 30° to the horizontal. A body of mass 2 kg is
released from the top of the slope and travels down against a constant resistance.
The body reaches the bottom of the slope with a speed of 2 m s-1.
Find the work done against the resistance and the magnitude of the resistance.
Advanced Higher
Mathematics of Mechanics
18)
A cyclist travels along a level road at a constant speed of 8 m s-1. If the resistances to motion total
50 newtons, find the rate at which the cyclist is working.
19)
A car of mass 900 kg is driven along a level road against a constant resistance to motion of
300 newtons. The maximum rate the engine can work is 12 kW.
Find the maximum velocity of the car on this level road.
Answers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
(5𝒊 + 12𝒋) m s-1, 13 m s-1
1·2 m s-1
4 newtons
1·48 m s-1
(−6𝒊 + 9𝒋) N s
8 m s-1, 4896 J, e.g. motion of bullet and gun follow the same linear path after firing
4 m s-1
(0𝒊 + 0𝒋) m s-1, i.e. T is stationary after the collision.
14 m s-1, e.g. air resistance is negligible
99 J
4
m s-1
3
6 m s-1
18·4 kW
147 J
14 m
1625 m
45 J, 9 N
400 W
40 m s-1
Advanced Higher
Mathematics of Mechanics
Force, Energy & Periodic Motion – Preparation for Assessment
Assessment Standard 1.2: Applying skills to motion in a horizontal circle with uniform angular velocity
1)
A satellite moves in a circular orbit round the Earth in the plane of the equator. The magnitude of
the gravitational force per unit mass that it experiences is 7.6 m s-2.
Given that the radius of the Earth is 6400 kilometres, calculate the height of the satellite above the
surface of the Earth.
2)
A satellite moves in a circular orbit around a planet in the plane of its equator and at a height of
600 km above the surface of the planet. The magnitude of the gravitational force per unit mass at
the surface of the planet is 8.6 m s-2.
Given that the radius of the planet is 5600 km, calculate the magnitude of the gravitational force
per unit mass that the satellite is experiencing in its orbit.
3)
A satellite moves in a circular orbit of radius 6800 km around a planet in the plane of its equator.
The magnitude of the gravitational force per unit mass that it experiences is 6·8 m s-2.
Given that the magnitude of the gravitational force per unit mass at the surface of the planet is
8·2 m s-2, calculate the radius of the planet.
4)
A car travels along a horizontal road which is an arc of a circle of radius 125 m. The greatest speed
that the car can travel without slipping is 42 km h-1.
Find the coefficient of friction between the tyres and the surface of the road.
5)
A circular bend of radius 300 m is banked at such an angle that no frictional force is necessary for a
car to take the bend at a speed of 25 m s-1.
Calculate the angle of banking.
6)
A circular bend of radius 250 m is banked at an angle of 10° to the horizontal.
Calculate the maximum speed at which a car can take the bend with no tendency to side-slip.
7)
A particle of mass 300 g is attached to one end of a light inextensible sting of length 40 cm, the
other end of the string being fixed at a point O on a smooth horizontal surface.
If the particle describes circles, centre O, find the tension in the string when:
(a)
the speed of the particle is 2√2 m s-1.
(b)
the angular speed of the particle is rad s-1.
8)
A small box is placed on the surface of a horizontal disc at a point 5 cm from the centre of the disc.
2
The box is on the point of slipping when the disc rotates at 1 5 rev s-1.
Find the coefficient of friction between the box and the surface of the disc.
9)
A hollow circular cylinder of radius 2 m rotates at 7 rad s-1 about its axis of symmetry which is
vertical. A body rotates, without slipping, on the inner surface of the cylinder.
Given that the body is on the point of slipping down the cylinder find the coefficient of friction
between the body and the cylinder.
Advanced Higher
Mathematics of Mechanics
10)
A conical pendulum consists of a light inextensible string fixed at a point A which lies vertically
above a point O. A particle of mass 50 g is attached to the other end of the string and moves in a
circle of radius √3 m about O. The angle between AO and the string is 30°.
Find the tension in the string and the angular speed of the particle.
11)
A particle is attached to one end of a light inextensible string, the other end of which is fixed.
The particle is describing horizontal circles, of radius 0·4 m, with uniform speed 1·5 m s-1, in a
horizontal plane below the fixed point.
Calculate the angle the string makes with the downward vertical through the fixed point.
Answers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
867·5 km
7·0 m s-2
6192 km
1
9
12°
20·8 m s-1
(a) 6 N (b) 3N
0395
01
05658 N, 1807 rad s-1
29·9°
Advanced Higher
Mathematics of Mechanics
Force, Energy & Periodic Motion – Preparation for Assessment
Assessment Standard 1.3: Applying skills to Simple Harmonic Motion
1)
The maximum speed of a particle describing simple harmonic motion about a point O is 10 m s-1.
When the displacement of the particle from O is 40 cm its speed is 6 m s-1.
Find the period and the amplitude of the motion.
2)
The maximum acceleration of a particle describing simple harmonic motion about a point O is
32 m s-2. The amplitude of the motion is 50 cm. Find the period of the motion and the speed of the
particle when it is at a distance of 30 cm from O.
3)
The maximum speed of a particle describing simple harmonic motion about a point O is 10 m s-1.
The amplitude of the motion is 50 cm.
Find the period of the motion and the distance of the particle from O when the speed of the
particle is 8 m s-1.
4)
A particle moves with simple harmonic motion about a mean position O. The amplitude of the
𝜋
motion is 65 cm and the period of the motion is 4 seconds.
Find how far the particle is from O when its speed is 2 m s-1.
5)
A light elastic string of natural length 75 cm has one end fixed and a mass of 800 g freely suspended
from the other end.
Find the modulus of elasticity of the string if the total length of the string and the mass in its
equilibrium position is 95 cm.
6)
A light elastic spring has its upper end fixed at A and a body of mass 600 g attached to its other end
hangs vertically below point A. The modulus of the spring is 45g N and its natural length is 15 m.
Find the extension of spring when the body hangs in equilibrium.
7)
When an elastic string is stretched to a length of 18 cm the tension in the string is 14 Newtons.
If the modulus of the string is 28 Newtons, find the natural length of the string.
Answers:
1)
2)
3)
4)
5)
6)
7)
𝜋
10
𝜋
4
𝜋
s, 50 cm
s, 32 m s-1
s, 30 cm
60 cm
3g N (294 N)
20 cm
12 cm
10
Advanced Higher
Mathematics of Mechanics
Force, Energy & Periodic Motion – Preparation for Assessment
Assessment Standard 1.4: Applying skills to Centres of Mass
For each of the questions 1 to 3 find the moment (or the sum of the moments) about the point A of
the forces shown.
1.
2.
5N
1m
1m
1m
1m
2m
A 1m
1m
A
6N
3N
12N
4N
6N
6N
3.
2m
A 1m
12N
6N
1m
1m
6N
6N
Answers:
1)
2)
3)
4Nm clockwise
6Nm anticlockwise
6Nm anticlockwise
Advanced Higher
Mathematics of Mechanics
Questions 1 to 5 show uniform laminae. Find the coordinates of the centre of gravity in
each one. Each grid consists of unit squares.
1.
3.
2.
4.
5.
Advanced Higher
Mathematics of Mechanics
Answers:
1)
2)
3)
(2.6, 1.9)
(2.3, 1.4)
1
(1, 2 3)
4)
(1.4, 2.2)
5)
(3, 3)
Advanced Higher
Mathematics of Mechanics