Work, energy and power

A particle P of mass 4 kg is moving in a straight line with a speed of 7 m s^-1 on a smooth horizontal surface.

A constant force of magnitude 12 N acts on P for 5 seconds in the same direction as P ’s motion.

Find the magnitude of the impulse exerted by the force on P. You must include the units.

Hence determine the speed of P at the instant when the force stops acting.

By considering the change in the kinetic energy of P, determine the work done by the force

over the 5 seconds that it acts.

Hence find the distance travelled by P while the force is acting on it.

In this question you must show detailed reasoning.

A particle P of mass 3.25 kg is moving on a smooth horizontal plane with velocity v ms^-1 where v = 4 i - 6 j.

Find the kinetic energy of P.

A system of horizontal forces starts to act on P. One of the forces is F N where F = i - (3 + t) j and t is the time, in seconds, after the system of forces starts to act.

Find the power developed by F at time t = 0.

Given that the velocity of P does not change at any time, find the total power developed by

the forces other than F at time t = 6.

One end of a light inextensible string of length r m is attached to a fixed point O. The other end is attached to a particle P of mass m kg.

Particle P hangs in equilibrium vertically below O with the string taut. An initial horizontal impulse is applied to P so that it instantaneously starts moving horizontally with a speed of u m s^-1.

Particle P subsequently moves in complete circles in a vertical plane containing O.

When the line OP makes an angle of θ with the downward vertical through O, the speed of P is v m s^-1 and the tension in the string is TN.

Show that T = u² + α gr + β gr cos θ) where α and β are constants to be determined.

Determine, in terms of m, g and r, the minimum initial impulse required to ensure that P

describes complete vertical circles around O with the string remaining taut.

One end of a light elastic string of natural length 1.4 m and modulus of elasticity 20 N is attached to a small object B of mass 2.5 kg. The other end of the string is attached to a fixed point O.

Object B is projected vertically upwards from O with a speed of u ms^-1.

State one assumption required to model the motion of B.

The greatest height above O achieved by B is 8.1 m.

Determine the value of u.

The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW. In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant.

At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is 25ms^-1.

Find the value of the resistance to motion.

The truck is being driven along the horizontal road with the engine working at 60 kW.

Find the acceleration of the truck at the instant when its speed is 10 ms^-1.

The truck is now driven down a straight road which is inclined at an angle θ below the horizontal. The greatest constant speed that the truck can be driven at maximum power is 40 ms^-1.

Determine the value of θ.

Find the tangential acceleration of P at A, stating its direction.

Determine the radial acceleration of P at A, stating its direction.

Find the magnitude of the force in the rod when P is at A, stating whether the rod is in tension

or compression.

The motion is now stopped when P is at A, and P is then projected in such a way that it now describes horizontal circles at a constant speed with θ = 40 ° (see diagram).

Find the speed of P.

Explain why, wherever P ’s motion is initiated from and whatever its initial velocity, it is not

possible for P to describe horizontal circles at constant speed with θ = 9 0 °.

Show that B’s motion can be modelled by the differential equation =-4.

Solve the differential equation in part (a) to find the particular solution for v in terms of x

and u.

By considering the behaviour of v as x → ∞ describe one feature of the model that is not

realistic.

At the instant when B reaches the point A, where x = X, its speed is V ms^-1. The work done by the resistance as B moves from O to A is denoted by W J.

(c)

Explain the relevance of the sign of your answer in part (c)(i).

By writing your answer to part (c)(i) in terms of V and u show how the quantity W

relates to the energy of B.

Show that at this instant the speed of P is 4.5 ms^-1.

Find the magnitude and direction of the radial acceleration of P at this instant.

Find the magnitude of the tangential acceleration of P at this instant.

A particle P of mass 3 kg is moving on a smooth horizontal surface under the influence of a variable horizontal force F N. At time t seconds, where t ⩾ 0, the velocity of P, v ms^-1, is given by

v = (32 sinh (2 t)) i + (32 cosh (2 t) - 257) j.

By considering kinetic energy, determine the work done by F over the interval

0 ⩽ t ⩽ ln 2.

Explain the significance of the sign of the answer to part (a)(i).

Determine the rate at which F is working at the instant when P is moving parallel to the

i-direction.

One end of a light elastic string of natural length 2.1 m and modulus of elasticity 4.8 N is attached to a particle, P, of mass 1.75 kg. The other end of the string is attached to a fixed point, O, which is on a rough inclined plane. The angle between the plane and the horizontal is θ where sin θ = The coefficient of friction between P and the plane is 0.732.

Particle P is placed on the plane at O and then projected down a line of greatest slope of the plane with an initial speed of 2.4 ms^-1.

Determine the distance that P has travelled from O at the instant when it first comes to rest. You

can assume that during its motion P does not reach the bottom of the inclined plane.

A car has mass 1200 kg. The total resistance to the car’s motion is constant and equal to 250 N.

The car is driven along a straight horizontal road with its engine working at 10 kW.

Find the acceleration of the car at the instant that its speed is 5 m s^-1.

The maximum power that the car’s engine can generate is 20 kW.

Find the greatest constant speed at which the car can be driven along a straight horizontal

road.

The car is driven up a straight road which is inclined at an angle θ above the horizontal where sin θ = 0.05.

Find the greatest constant speed at which the car can be driven up this road.

The coordinates of two points, A and B, are (- 1, 6) and (5, 12) respectively, where the units of the coordinate axes are metres. A particle P moves from A to B under the action of several forces. The force F = 7 i - 2 j N is one of the forces acting on P.

Calculate the work done by F on P as P moves from A to B.

At the instant when P reaches B its velocity is -i - 5j m s^-1.

Find the power generated by F at the instant that P reaches B.

One end of a light elastic string was attached to the origin of the coordinate system and the other to P when P was at A, before it moved to B. The natural length of the string is 8 m and its modulus of elasticity is 24 N.

At the instant that P reaches B, find the following.

• The tension in the string

• The elastic potential energy stored in the string

A particle P of mass 6 kg moves in a straight line under the action of a single force of magnitude F N which acts in the direction of motion of P. At time t seconds, where t ⩾ 0, F is given by F = When t = 0, the speed of P is 1.9 m s^-1.

Find the impulse of the force over the period 0 ⩽ t ⩽ 2.

Find the speed of P at the instant when t = 2.

Find the work done by the force on P over the period 0 ⩽ t ⩽ 2.

The training rig for a parachutist comprises a fixed platform and a fixed hook, H. The platform is 3.5 m above horizontal ground level. The hook, which is not directly above the platform, is 6.5 m above the ground.

One end of a light inextensible cord of length 4.5 m is attached to H and the other is attached to a trainee parachutist of mass 90 kg standing on the edge of the platform with the cord straight and taut.

The trainee is then projected off the platform with a velocity of 7 m s^-1 perpendicular to the cord in a downward direction. The motion of the trainee all takes place in a single vertical plane and while the cord is attached to H it remains straight and taut.

When the speed of the trainee reaches 5.5 m s^-1 the cord is detached from H and the trainee then moves under the influence of gravity alone until landing on the ground (see diagram).

The trainee is modelled as a particle and air resistance is modelled as being negligible.

Show that at the instant before the cord is detached from H, the tension in the cord has a

magnitude of 1005.5 N.

The point on the ground vertically below the edge of the platform is denoted by O. The point on the ground where the trainee lands is denoted by T.

Determine the distance OT.

The ground around T is in fact an elastic mat of thickness 0.5 m which is angled so that it is perpendicular to the direction of motion of the trainee on landing. The mat, which is very rough, is modelled as an elastic spring of natural length 0.5 m. It is assumed that the trainee strikes the mat at ground level and is brought to rest once the mat has been compressed by 0.3 m.

Determine the modulus of elasticity of the mat. Give your answer to the nearest integer.

Calculate the work done by the force during these 5 seconds.

Calculate the average power generated by the force during these 5 seconds.

The speed of P at O is 10ms^-1.

Calculate the speed of P at A.

A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N. The rope has natural length 4 m and modulus of elasticity 470 N.

By considering energy, determine the total distance she falls before first coming to instantaneous

rest.

Show that v² = 50.7 - 14.7 cos θ.

Given that the string breaks when the tension in it reaches 200 N, find the angle that OA turns

through between the instant that A is projected and the instant that the string breaks.

Two smooth circular discs A and B are moving on a horizontal plane. The masses of A and B are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of A is 2ms^-1 at an angle of 60° to the line joining their centres,

• the velocity of B is 5ms^-1 towards A along the line joining their centres (see Fig. 6).

Given that the velocity of A after the collision is perpendicular to the velocity of A before the collision, find

the coefficient of restitution between A and B,

the total loss of kinetic energy as a result of the collision.