Newton’s laws of motion

A car moves in a straight line along a horizontal road with constant acceleration 2 m s^–2 The car is moving with speed 15 m s^–1 in the direction of the acceleration when it passes a signpost on the road. The car is modelled as a particle.

Use the model to find the speed of the car 4 s after passing the signpost. Figure 1 below shows the horizontal forces acting on the car. Given that

• the car has mass 800 kg

• the driving force of the engine has magnitude D newtons

• the resistance to the motion of the car has magnitude 400 N

• the acceleration of the car is 2 m s^–2 in the direction of the driving force

use the model to find the value of D.

A small box B of mass 2 kg is dragged in a straight line, along a rough horizontal plane, at a constant speed by a force of magnitude 5 N. The line of action of the force makes an angle α with the plane, where sinα = as shown in Figure 2.

Show that the magnitude of the normal reaction of the plane on the box is 16.6 N. At the instant when B is at the point O on the plane, the force of magnitude 5 N is removed.

Describe the motion of the box after the force of magnitude 5 N is removed.

Find the magnitude of the normal reaction of the plane on the box after the force of magnitude 5 N is removed. Given that after the force of magnitude 5 N is removed

• the box is modelled as a particle

• air resistance is modelled as being negligible

• the coefficient of friction between the box and the plane is modelled as 0.2

• the speed of the box as it passes through O is 4 m s^–1

• the box comes to rest at the point X on the plane

use the model to find the length OX.

State one limitation of the model, apart from ignoring air resistance, that could affect your answer to part (d).

[In this question, i and j are horizontal unit vectors due east and due north respectively.] A particle P of mass 0.5 kg moves with constant acceleration (2i – 2.4j) m s^–2 on a smooth horizontal plane under the action of a constant horizontal force F N.

Find F in terms of i and j. At time t = 0, P is moving with velocity (–7i + 7.8j) m s^–1

Find the velocity of P at time t = 2 seconds.

Find the direction of motion of P at time t = 2 seconds, giving your answer as a bearing in degrees. At time t = 0, P passes through the point O. At time t = 5 seconds, P passes through the point A.

Find OA → in terms of i and j.

A particle P of mass m is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle α, where tanα =

• the coefficient of friction between P and the plane is μ, whereμ

The particle P is released from rest and slides down the plane. Air resistance is modelled as being negligible. Using the model,

find, in terms of m and g, the magnitude of the normal reaction of the plane on P,

show that, as P slides down the plane, the acceleration of P down the plane is

State what would happen to P if it is released from rest butμ

A particle P has mass 5 kg. The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N. The only resistance to motion is a frictional force of magnitude F newtons, as shown in Figure 1.

Find the magnitude of the normal reaction of the plane on P The particle is accelerating along the plane at 1.4 m s^–2

Find the value of F The coefficient of friction between P and the plane is μ

Find the value of μ, giving your answer to 2 significant figures.

A small stone A of mass 3m is attached to one end of a string.

A small stone B of mass m is attached to the other end of the string.

Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα =

The string passes over a pulley P that is fixed at the top of the plane.

The part of the string from A to P is parallel to a line of greatest slope of the plane.

Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is

Stone A is released from rest and begins to move down the plane.

The stones are modelled as particles.

The pulley is modelled as being small and smooth.

The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer.

In reality, the string is not light.

State how this would affect the working in part (b).

A small stone A of mass 3m is attached to one end of a string. A small stone B of mass m is attached to the other end of the string. Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα = The string passes over a pulley P that is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is Stone A is released from rest and begins to move down the plane. The stones are modelled as particles. The pulley is modelled as being small and smooth. The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer. In reality, the string is not light.

State how this would affect the working in part (b).

A lift of mass 250 kg is being raised by a vertical cable attached to the top of the lift. A woman of mass 60 kg stands on the horizontal floor inside the lift, as shown in Figure 3. The lift ascends vertically with constant acceleration 2 m s^–2. There is a constant downwards resistance of magnitude 100 N on the lift. By modelling the woman as a particle,

find the magnitude of the normal reaction exerted by the floor of the lift on the woman.

The tension in the cable must not exceed 10 000 N for safety reasons, and the maximum upward acceleration of the lift is 3 m s^–2. A typical occupant of the lift is modelled as a particle of mass 75 kg and the cable is modelled as a light inextensible string. There is still a constant downwards resistance of magnitude 100 N on the lift.

Find the maximum number of typical occupants that can be safely carried in the lift when it is ascending with an acceleration of 3 m s^–2.

[ In this question i and j are horizontal unit vectors due east and due north respectively ] Two forces F₁ and F₂ act on a particle P of mass 0.5 kg. F₁ = (4i – 6j) N and F₂ = (pi + qj) N. Given that the resultant force of F₁ and F₂ is in the same direction as –2i – j,

show that p – 2 q = –16

Given that q = 3

find the magnitude of the acceleration of P,

find the direction of the acceleration of P, giving your answer as a bearing to the nearest degree.

A vertical rope AB has its end B attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg, as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration 0.5 m s^–2 using the rope AB. The rope is modelled as a light inextensible string.

Find the tension in the rope AB.

Find the magnitude of the force exerted on the scale pan by the brick.

Two forces F₁ and F₂ act on a particle P. The force F₁ is given by F₁ = (–i + 2j) N and F₂ acts in the direction of the vector (i + j). Given that the resultant of F₁ and F₂ acts in the direction of the vector (i + 3j),

find F₂

The acceleration of P is (3i + 9j) m s^–2. At time t = 0, the velocity of P is (3i – 22j) m s^–1

Find the speed of P when t = 3 seconds.

Two forces (4 i – 2 j) N and (2 i + q j) N act on a particle P of mass 1.5 kg. The resultant of these two forces is parallel to the vector (2 i + j).

Find the value of q.

At time t = 0, P is moving with velocity (–2i + 4j)m s^–1.

Find the speed of P at time t = 2 seconds.

A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg. The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of 2 m s^–2. By modelling the cable as being light and inextensible, find

the tension in the cable,

the magnitude of the force exerted on the woman by the floor of the lift.

A particle P is projected vertically upwards from a point A with speed u m s^–1. The point A is 17.5 m above horizontal ground. The particle P moves freely under gravity until it reaches the ground with speed 28 m s^–1.

Show that u = 21

At time t seconds after projection, P is 19 m above A.

Find the possible values of t.

The ground is soft and, after P reaches the ground, P sinks vertically downwards into the ground before coming to rest. The mass of P is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on P.

Find the vertical distance that P sinks into the ground before coming to rest.

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle θ, where tan , and released from rest. The coefficient of friction between the brick and the plane is Find the acceleration of the brick.

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s^–2. Find

the angle between the acceleration and i,

the magnitude of F.

At time t seconds the velocity of P is v m s^–1. Given that when t = 0, v = 9i – 10j,

find the velocity of P when t = 5.

A particle P of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of 30 to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N, acting at an angle to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.

Find the normal reaction between P and the plane.

The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to P so that P moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.

Find the initial acceleration of P.