Solved problems for the chapter:

Newton's Laws of Motion

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Force and Acceleration

1. Newton's second law or law of dynamics

2. Newton's first law or inertia of the body

3. The force of gravity

4. Forum

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Introduction to Force

1. What is force?

2. How do we know that a force is acting on a body?

2.1. Effects of forces

3. Name, symbol and unit of force

4. Properties of force

5. Drawing forces

6. Contact force or field force

7. Surface and volume distributed forces

7.1. Area-distributed or surface forces

7.2. Volume-distributed or body forces

8. Measuring forces

9. Forum

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Newton's Third Law of Motion

1. Newton's third law

2. Internal forces

3. Hooke's law

3.1. Spring technique

4. Forum

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Newton's Second Law of Motion

1. Force, mass, and acceleration

1.1. A force acting in the direction of motion

1.2. A force acting opposite to the direction of motion

2. Weight

2.1. A body resting on a horizontal surface

2.2. A body falling freely

2.3. A body moving down a slope

3. Forum

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Momentum

1. Impulse - momentum theorem

2. Law of conservation of momentum

2.1. Elastic and Inelastic collisions

2.1.1. Elastic Collision

2.1.2. Inelastic collision

2.1.3. Partially elastic collision

2.2. External and internal force impulses

2.3. Law of conservation of momentum

3. Forum

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Newton's First Law of Motion for KS4

1. Newton's first law

2. Equilibrium of forces

2.1. Equilibrium of two forces

2.2. Equilibrium of several forces

2.2.1. Parallelogram rule

2.2.2. Polygon forces

3. Forum

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Addition of Forces for KS4

1. Resultant forces

2. Addition of parallel forces

2.1. Addition of forces acting in the same direction

2.2. Addition of forces acting in the opposite direction

2.2.1. Equal and opposite forces

3. Addition of non-parallel forces

3.1. Parallelogram rule

3.2. Polygon of forces

4. Forum

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Kinetic Energy for A-Levels

1. Translational kinetic energy

1.1. Work - Kinetic energy theorem

2. Rotational kinetic energy

2.1. Moment of inertia of a real body

2.2. Kinetic energy of a rolling body

3. Forum

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:newton_039%3Bs_laws_of_motion/5078/

Example 1.

A stationary trolley with mass is pushed through a distance of in 20 seconds. How much force is applied on the trolley?

Example 2.

A car with a mass of drives on a flat road at a speed of . How much braking force do we need to stop it after traveling a distance of ?

An elevator with a mass of starts to rise with an acceleration of . What is the force in the elevator rope? What will be the force in the rope if the elevator descends with the same acceleration?

Two bodies are suspended via a rope on an attached pulley. The first body has a mass of 0.1 kg and the second 0.3 kg. The bodies are first held with the hand and then released. With what acceleration are they moving? What is the force on both sides of the rope?

A body slides on a rough surface and stops at a distance of . The coefficient of friction between the body and the base is . What is the initial velocity of the body?

A body slides at a speed of on a smooth horizontal surface. What is the distance covered after a coefficient of friction of 0.2 occurs between the body and the ground?

A rocket model with a mass of is launched vertically upwards with an initial velocity of . It reaches its maximum height in . With what average force does air resistance impede the rocket?

A stationary elevator with a mass of rises through a height of in . How much force does the elevator rise with?

The position of a body in a plane is described by the equations:

In this case, is the time in seconds, and and are the positions of the body in the coordinate system in metres. The coordinate is the altitude. What's the magnitude of the forces and acting on the body with weight ?

10.

Two bodies with masses and lying on a horizontal smooth surface are connected by a light rope. The rope breaks at a force greater than . Two forces and begin to act on the bodies in opposite directions as shown in the figure below. The constant , and is the time in seconds.

Body on a horizontal smooth surface .

Obtain an equation that shows the change in the acceleration of the bodies with time.
What is the force in the rope after from the start of the forces?
How long does it take for the rope to break?

11.

A brick slips from the top of a roof as shown in the figure below. The roof is long and is inclined at an angle to the horizontal. The height of the outer wall of the house is and the coefficient of friction between the brick and the roof is .

Brick on the roof

What external forces act on the brick? Draw the forces and name them.
With what acceleration does the brick slide on the roof?
With what speed does the brick slide off the roof?
How long does it take the brick to fall to the ground after slipping off the roof?
Where does the brick fall to the ground?
With what speed and at what angle with respect to the horizontal does the brick fall to the ground?

12.

A car is driven into a sharp bend in a horizontal road. The radius of the bend is , and the coefficient of friction between the tires and the ground is .

Driving a car in a bend .

Draw the external forces on the car and name them.
What does the maximum speed of a car depend on when driving safely through a bend in a horizontal road?
What is the maximum speed with which the car can drive safely around the bend?
On normal road traffic, only the right half of the road should be used. Based on the equation obtained in part b above, explain on which side of the bend the car can drive at a higher speed.

13.

Two bodies of the same mass are attached to the ends of a light and inextensible rope, which is passed over a pulley. A smaller body of a mass is attached to the body on the right side of the rope. The mass of the rope and the pulley and the friction in the pulley are neglected.

Weights on a string .

Draw the forces on the body on the left and on the system of bodies on the right side of the rope and name them.
With what acceleration does the whole system start to move?
What is the tension force in the rope?
Calculate the force with which the body of mass acts on the body of mass .

14.

Two logs with masses and that are connected by a rope that is passed over a light pulley lie on a slope as shown in figure 1 below:

Figure 1: Connected logs on slopes

The slopes are inclined at angles and to the horizontal. The coefficient of friction between the logs and the slope is . The log with mass is held above the second log with mass and both are released at some point.

What is the distance the logs will have to travel for them to be at the same height?
What is the acceleration of the logs?
How long does it take the logs to be at the same height after they start moving?
What is the speed of the logs at that time?

15.

Logs of masses and are connected by a rope which is passed over a light pulley as shown in figure 1 below.

Figure 1: Connected logs on slopes .

The angles of the inclined planes with respect to the horizontal are and . The log with mass is lower than the log with mass . After the logs move for 3 seconds, they are both at the same height. The coefficient of friction between the logs and the plane is . What is the ratio of the mass to that of ?

16.

With the help of Newton's second law, explain the following phenomena:

We are standing in a bus when it suddenly starts. Why can we lose our balance? In which direction can we fall if we face the direction of travel? How do we prevent this?
A weight is hanging on a rope attached to the ceiling of a car. In which direction will it deviate during acceleration and in which direction during braking?
We are sitting in a car. We have a container of water in our hands. We observe the water level. How do we know when a car is accelerating, braking, or driving steadily?
We lower a 15 kg sand container uniformly with a rope. How much force do we feel in our hands? Do we feel more or less force while dropping the container on the ground?
Explain how the seat belt works during a sudden stop. What does the force with which the belt acts on the driver when it stops depend on?
When a plane accelerates along the runway we feel it sticking us to the backrest. Explain the phenomenon.

17.

Kelvin keeps a barrel on an inclined plane so that it doesn't roll down. The angle of the slope is and the weight of the barrel is 80 kg. With what force does Kelvin hold on to the barrel?

18.

A car weighing 1500 kg reaches a speed of 24 m/s in 8 s. What is the resultant of the forces acting on the car?

19.

On a 30 kg trolley, a force of 60 N is applied for 6 s. What is the speed of the trolley?

20.

A log weighing 50 kg is dragged on a horizontal surface. The frictional force between the log and the surface is 200 N. What force must be applied on the log for the acceleration to be ?

21.

Two trolleys weighing 40 kg and 60 kg are connected by a drawbar. We pull the heavier trolley with a force of 200 N.

What is the acceleration of the system if we ignore the friction?
How much force does the drawbar apply on the lighter trolley?

22.

A trolley with a mass of 0.9 kg is tied with a rope via a fixed pulley to a weight of 0.1 kg as shown in the figure below.

How fast do the bodies move?
What is the force with which the rope accelerates the trolley?

23.

In a car, a passenger weighing 70 kg is fastened with a seat belt. The car is traveling at a speed of 36 km/h when it suddenly stops due to a collision. The time it takes to stop is 0.2 s. How strong is the seat belt holding the passenger?

24.

On an aircraft carrier, a special device called a catapult accelerates a military aircraft to take-off speed. On a 90 m long runway, the aircraft reaches a takeoff speed of 100 m/s. With what force does the catapult act on the aircraft? The mass of the aircraft is 45 t.

25.

The graph of the acceleration of a body with a mass of 5 kg against the time is as shown below. Draw the graph of the force acting on the body against the time.

Acceleration-time graph of the body

26.

Identify the correct and the incorrect statements in the list below:

A body moves with uniform acceleration if the force on the body increases uniformly with time.
A body moves with uniform deceleration if a constant force acts on the body in the opposite direction to its motion.
If the mass of a body is reduced twice at the same force, the acceleration is also reduced twice.
If the force acting on a body is ten times greater, the acceleration is also ten times greater.
If a ship is moving uniformly and in a straight line, a constant force is acting on it.
If a constant force acts on a body in the direction of motion, the body accelerates uniformly. If the force acts in the opposite direction, it decelerates uniformly.
If the force on a body changes over time, the motion is non-uniform.
If a car goes into a corner at a uniform speed, the resultant of the forces acting on the car is zero.
A stone is thrown vertically upwards. When it reaches its maximum height, just before it begins to fall down, it stops for a moment. At this point, the resultant of the forces acting on the stone is zero.

27.

A body weighing 60 kg moves non-uniformly. The figure below is a graph of its velocity against time. Draw the graph of the resultant forces on the body against time.

Time-dependent velocity graph

28.

A mass of 6 kg is pulled with two equally large forces of magnitude 8 N without friction on a smooth surface. Both forces act at an angle with respect to the direction of motion, as shown in the figure below. How much acceleration does the body move with?

29.

A body weighing 600 g is on a slope at an angle of to the horizontal. The force of friction between the body and the slope is 0.5 N.

At what acceleration does the body move?
Use the graphical method to determine the forces on the slope.

Body on the slope

30.

Two masses of 0.5 kg each are suspended on the rope of an attached pulley - each on its own side (Figure A). Are the masses at rest? What is the force of the rope?
We hang an additional weight with a mass of 0.1 kg on the left side (picture B). In which direction is the weight moving? What is the acceleration of the motion?

31.

A log with a mass of 2 kg is dragged on a flat surface with a force of 20 N and at an angle with respect to the horizontal. The log moves with an acceleration of . What is the force of friction between the log and the surface?

A log dragged on a flat horizontal surface

32.

The rocket for supplying the International Space Station has a mass of 240 t with cargo at takeoff. The thrust force of the rocket engines is 3.2 MN. With what acceleration does it rise at takeoff?

33.

A 560 t aircraft has two engines. Each generates a thrust of 600 kN. At what time does the aircraft reach a take-off speed of 300 km/h?

34.

A rope attached to the ceiling hangs a 2 kg weight. A second rope is attached to this weight, on which hangs a 5 kg weight. What are the tension forces on both ropes and where are they directed?

35.

An elevator with a mass of 800 kg rises uniformly with an acceleration of . What is the force of the elevator rope?

36.

Two skaters are holding a rope in their hands as shown in the picture. The weight of the first skater is 40 kg and the second 60 kg. The first skater pulls the second with a force of 60 N. What are the speeds of both skaters after two seconds of pulling? Friction is ignored.

37.

Are the following statements correct?

If we act on a body with a certain force, the body also acts on us with an equal and opposite force.
Bodies interact only with direct contact.
A body acts on another body with a force F, and the second body on the first with the opposite same force. Since the sum of both forces is zero, the body is at rest or moving evenly and in a straight line.
The law of interaction also applies when a body acts with force on another body and accelerates it.
The earth acts on the Moon with the same force as the Moon on the Earth.

38.

Determine whether the description of Newton's first or third laws applies in the given cases.

A car drives uniformly on a level road. It is acted upon by the force of the engine and, conversely, by the same force of air resistance.
The force of water resistance inhibits the navigation of the ship, and the ship acts with force on the water to cause waves. Both forces are equal and opposite.
Push a trolley on a flat surface with the force of your hands so that it moves uniformly.
A locomotive acts with force F on the train carriages and accelerates them. The wagons operate with the same opposite force on the locomotive.
A body hangs on a string. The sum of the force on the string and the weight of the body is zero, so the body is at rest.
A body hangs on a string. With its weight it acts on the string vertically downwards, and the string with the equal and opposite force on the body.

39.

Identify which of the listed motions below is described by Newton's first law. Justify the answer.

Brock meandering evenly along the snowy slope.
Raindrops falling from the sky at a speed of 8 m/s.
Children on the carousel rotating at a constant speed.
A passenger plane flying on a straight track and at a constant speed over the Atlantic.
A stone falling to the ground from a height of 10 m.

40.

Tom pulls a log with a force of 30 N at an angle of 45 ° with respect to the horizontal. The frictional force between the log and the ground is 21 N, the weight of the log is 40 N, and the ground acts with a force of 19 N on the log. Draw all the forces acting on the log on a scale and determine whether Newton's first law applies to the motion of the log.

41.

A car is driven on a flat road. The force of the engine is constantly opposed by the force of air resistance, which depends on the speed of the vehicle. Consider four cases:

The engine force is equal to the resistance force. The magnitude of each is 0 N.
The engine force is 800 N and the resistance force is 200 N.
The engine force is equal to the resistance force. The magnitude of each is 400 N.
The engine force is 200 N and the resistance force is 400 N.

Which of the cases does Newton's first law describe?

42.

A square-shaped log is dragged across a table so that it moves uniformly and on a straight line. If the applied force is 4 N and the weight of the log is 6 N, draw to scale, all the forces acting on the log.

43.

Karl and John are standing on skateboards. John has a rope tied around his waist, with which Karl pulls him towards himself with a force of 30 N. Karl notices that during the pulling, he also started moving towards Jan. Explain why.

44.

The mass of a rocket filled with fuel is 380 tonnes. The rocket engines develop a thrust of 4.6 MN. Calculate the resultant of the forces acting on the rocket after take-off.

45.

A pan containing water is placed on a cooker. It acts on the cooker plate with a force of 40 N. Name the force and determine its direction and line of action. Draw the force to scale.

46.

A man pulls an 800 kg log vertically upwards with a force of 6 kN. Will he be able to lift it? What is the resultant force of the log on the ground during the lifting?

47.

Joseph holds a bucket of mortar tied to a rope so that the bucket rests. The rope is passed over a pulley attached to the top of the masonry scaffold, as shown in the figure. On Joseph's side, he pulls the rope at an angle of to the vertical. The weight of the bucket with mortar is 20 kg. How much force does Joseph apply on the rope? Does the magnitude of the force depend on the angle at which Joseph pulls the rope?

48.

Victor wants to lift a block of mass 120 kg using a pulley. The pulley has a fixed part with two wheels and a moving part with one wheel. A tow rope is passed through all three wheels, as shown in the figure. He hangs the load on the moving part of the pulley. What force does Victor apply on the free part of the rope to keep the load in the air at rest? The moving part of the pulley has a negligible mass.

49.

A crate of mass 80 kg rests strapped to the rope of a fixed and movable pulley as shown in the figure. With how much force does Vincet have to pull the rope to keep the crate stationary in the air?

50.

A load of mass 40 kg rests a smooth inclined plane of angle to the horizontal. The block is attached to an unknown weight with a rope over a pulley so that it does not slip down the slope - see picture. What is the mass of the unknown weight?

51.

A professor is standing on the stage. His weight is 700 N. With what force does the stage act on the professor?

52.

Zane and Kelvin stand on skateboards and push off with their hands. Kelvin acts on Zane with a force of 40 N. With what force does Zane act on Kelvin? How will they move after the push?

53.

A ball falls to the ground. Describe this using Newton's 3rd law, a pair of equal and opposite forces acting on the ball.

54.

A spring balance hangs from the ceiling. A ball of mass 0.2 kg is hung on it. At the bottom of the ball is another spring balance, through which we act vertically downwards with a force of 4 N. What force does the upper spring balance show?

55.

A lightweight spring balance hangs on a stand with a 2 N weight suspended on it. Determine at least three pairs of forces according to Newton's third law acting on the stand, spring, and weight.

56.

A car accelerates on a flat road. The motion of the car is constantly inhibited by the force of air resistance. Describe what applies to air according to Newton's 3rd law.

57.

A log placed on a slope with an inclination of to the ground has a weight of 80 N. With what force does the slope act on the log?

58.

A box is lying on the table. The box is pushed by a finger with a force of 6 N. The box then begins to move. The motion of the box is hindered by a frictional force of 4 N. With what force does the box act on the finger? Does this force depend on the frictional force?

59.

A magnet attracts an iron box with a force of 6 N.

With what force does the box attract the magnet?

60.

A boat sails uniformly over the sea. It is subjected to an engine force of 20 kN.

What is the force of water resistance?
Does the boat also have a force on the water?
How do you know?

61.

On a slope with an inclination of , John holds a barrel with a force of 300 N with his hand. The dynamic component of the force on the slope is 400 N. How will the barrel move? How much force will the barrel exert on John's hand?

62.

Snowflakes fall uniformly at a speed of 2 m/s. The weight of a snowflake is 0.1 g. What forces act on a snowflake and what is their size?

63.

A cyclist is riding at a steady speed on a level road. According to Newton's third law of motion, what is equal and opposite to the backward push of the back wheel on the road?

the force exerted by the cyclist on the pedals
the forward push of the road on the back wheel
the tension in the cycle chain
the total air resistance and friction force

64.

A force is applied to a freely moving object. At one instant of time, the object has velocity and

acceleration . Which quantities must be in the same direction?

and only
and only
and only
, and

65.

Which is not one of Newton's laws of motion?

The total momentum of a system of interacting bodies remains constant, providing no external force acts.
The rate of change of momentum of a body is directly proportional to the external force acting on the body and takes place in the direction of the force.
If body A exerts a force on body B, then body B exerts an equal and oppositely-directed force on body A.
A body continues in a state of rest or of uniform motion in a straight line unless acted upon by some external force.

66.

A box is shown resting on the ground. Newton’s third law implies that four forces of equal magnitude are involved. These forces are labelled P, Q, R and S. Forces P and Q act on the box. Forces R and S act on the Earth. For clarity, the forces are shown slightly separated.

Which statement about the forces is correct?

P is the equal and opposite force to Q and both are forces of contact.
Q is the equal and opposite force to P and both are gravitational forces.
R is the equal and opposite force to S and both are forces of contact.
S is the equal and opposite force to Q and both are gravitational forces.

67.

Which statement about Newton’s laws of motion is correct?

The first law follows from the second law.
The third law follows from the second law.
Conservation of energy is a consequence of the third law.
Conservation of linear momentum is a consequence of the first law.

68.

A rocket of mass sits on a launch pad on the Earth’s surface. The rocket motors provide an upward force of on the rocket.

What is the initial acceleration of the rocket?

69.

What is the definition of the force on a body?

the mass of the body multiplied by its acceleration
the power input to the body divided by its velocity
the rate of change of momentum of the body
the work done on the body divided by its displacement

70.

State Newton’s second law.
A ball of mass hits a wall with a velocity of perpendicular to the wall. The ball rebounds perpendicularly from the wall with a speed of . The contact time of the ball with the wall is .

Calculate, for the ball hitting the wall,
- the change in momentum,
- the magnitude of the average force.

71.

Newton’s third law of motion is often summarized as ‘Every action (force) has an equal and opposite reaction.’ A book rests on a table. If the weight of the book is the ‘action’ force, what is the ‘reaction’ force?

the pull of the book on the Earth
the pull of the Earth on the book
the push of the book on the table
the push of the table on the book

72.

A ball of mass is at rest on a horizontal floor.

- Calculate the magnitude of the normal contact force from the floor acting on the ball.
- Explain your working in (i).
The ball is now lifted above the floor and dropped so that it falls vertically,

Just before contact with the floor, the ball has velocity downwards. The ball bounces from the floor and its velocity just after losing contact with the floor is upwards. The ball is in contact with the floor for .
- State Newton’s second law of motion.
- Calculate the average resultant force on the ball when it is in contact with the floor.

73.

A light rope is installed over a lightly attached pulley. On the rope are hung bodies with masses and as shown in the figure below. Initially, the bodies are held so that their centers of gravity are apart. Ignore friction, if the bodies are released:

how fast does the heavier body fall?
how long does it take for the centers of gravity of the bodies to be at the same height?
what is the velocity of each body at this height?
What is the kinetic energy of the body system when the centers of gravity of the bodies are at the same height?

Figure 1: Bodies on a pulley