Connected particles

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 particle P of mass 4 m is held at rest at the point X on the surface of a rough inclined plane which is fixed to horizontal ground. The point X is a distance h from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle α where tan . The coefficient of friction between P and the plane is The particle P is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle Q of mass m which hangs freely at a distance d, where d h, below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and P moves down the plane. For the motion of the particles before P hits the ground,

state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

When P hits the ground, it immediately comes to rest. Given that Q comes to instantaneous rest before reaching the pulley,

A vertical light rod PQ has a particle of mass 0.5kg attached to it at P and a particle of mass 0.75kg attached to it at Q, to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15N applied to the particle at Q. Find the thrust in the rod.

Two particles, A and B, have masses 2 m and m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a fixed rough horizontal table at a distance d from a small smooth light pulley which is fixed at the edge of the table at the point P. The coefficient of friction between A and the table is , where < The string is parallel to the table from A to P and passes over the pulley. Particle B hangs freely at rest vertically below P with the string taut and at a height h, (h < d), above a horizontal floor, as shown in Figure 3. Particle A is released from rest with the string taut and slides along the table.

Write down an equation of motion for

Write down an equation of motion for

Hence show that, until B hits the floor, the acceleration of A is 1 2μ).

Find, in terms of g, h and μ, the speed of A at the instant when B hits the floor.

After B hits the floor, A continues to slide along the table. Given that — and that A comes to rest at P,

find d in terms of h.

Describe what would happen if

Two particles P and Q have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle P is held at rest on a fixed rough horizontal table. The coefficient of friction between P and the table is . The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle Q hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle P is released from rest with the string taut and slides along the table. Assuming that P has not reached the pulley, find

the tension in the string during the motion,

the magnitude and direction of the resultant force exerted on the pulley by the string.

Figure 2

Two particles A and B have masses 2 m and 3 m respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and A and B are above a horizontal plane, as shown in Figure 2. The system is released from rest.

Show that the tension in the string immediately after the particles are released

After descending 1.5 m, B strikes the plane and is immediately brought to rest. In the subsequent motion, A does not reach the pulley.

Find the distance travelled by A between the instant when B strikes the plane and the instant when the string next becomes taut.

Given that m = 0.5 kg,

find the magnitude of the impulse on B due to the impact with the plane.

A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley P is fixed at the top of the plane. Two particles A and B, of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley P. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs freely below P, as shown in Figure 2. The coefficient of friction between A and the plane is . Initially A is held at rest on the plane. The particles are released from rest with the string taut and A moves up the plane. Find the tension in the string immediately after the particles are released.

A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle to the road, as shown in Figure 4. The vehicles are travelling at 20 m s^–1 as they enter a zone where the speed limit is 14 m s^–1. The truck’s brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s^–1 is 100 m.

Find the deceleration of the truck and the car.

The constant braking force on the truck has magnitude R newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos = 0.9, find

the force in the towbar,

the value of R.

Two particles A and B have masses 2 m and 3 m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle B hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle A is released from rest. Assuming that A has not reached the pulley, find

the acceleration of B,

the tension in the string,

the magnitude and direction of the force exerted on the pulley by the string.

Figure 5 shows two particles A and B, of mass 2 m and 4 m respectively, connected by a light inextensible string. Initially A is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle , where tan . The coefficient of friction between A and the plane is . The string passes over a small smooth pulley P which 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 and B hangs vertically below P. The system is released from rest with the string taut, with A at the point X and with B at a height h above the ground. For the motion until B hits the ground,

give a reason why the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

Particle B does not rebound when it hits the ground and A continues moving up the plane towards P. Given that A comes to rest at the point Y, without reaching P,

find the distance XY in terms of h.

Two particles P and Q, of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time t = 0, a constant force F of magnitude 4 N is applied to Q in the direction PQ, as shown in Figure 3. The system moves under the action of this force until t = 6 s. During the motion, the resistance to the motion of P has constant magnitude 1 N and the resistance to the motion of Q has constant magnitude 2 N. Find

the acceleration of the particles as the system moves under the action of F,

the speed of the particles at t = 6 s,

the tension in the rod as the system moves under the action of F.

At t = 6 s, F is removed and the system decelerates to rest. The resistances to motion are unchanged. Find

the distance moved by P as the system decelerates,

the thrust in the rod as the system decelerates.

Two particles P and Q have masses 0.3 kg and m kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at

The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle P is held at rest on the inclined plane and the particle Q hangs freely below the pulley with the string taut, as shown in Figure 2.

Find

the magnitude of the normal reaction of the inclined plane on P,

the value of m.

When the particles have been moving for 0.5 s, the string breaks. Assuming that P does not reach the pulley,

find the further time that elapses until P comes to instantaneous rest.

Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially B is held at rest on a rough fixed plane inclined at angle to the horizontal, where tan = . The part of the string from B to P is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, P, fixed at the top of the plane. The particle A hangs freely below P, as shown in Figure 4. The coefficient of friction between B and the plane is . The particles are released from rest with the string taut and B moves up the plane.

Find the magnitude of the acceleration of B immediately after release.

Find the speed of B when it has moved 1 m up the plane.

When B has moved 1 m up the plane the string breaks. Given that in the subsequent motion B does not reach P,

find the time between the instants when the string breaks and when B comes to instantaneous rest.

Two particles A and B have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion B does not reach the pulley.

Find the tension in the string immediately after the particles are released.

Find the acceleration of A immediately after the particles are released.

When the particles have been moving for 0.5 s, the string breaks.

Find the further time that elapses until B hits the floor.

Two particles A and B have masses 5 m and km respectively, where k < 5. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with A and B at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, A descends with acceleration g.

Show that the tension in the string as A descends is mg.

Find the value of k.

State how you have used the information that the pulley is smooth.

After descending for 1.2 s, the particle A reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between B and the pulley is such that, in the subsequent motion, B does not reach the pulley.

Find the greatest height reached by B above the plane.

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

the acceleration of the car and trailer,

the magnitude of the tension in the towbar.

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude F newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

find the value of F.

Two particles P and Q, of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force F of magnitude 30 N is applied to Q in the direction PQ, as shown in Figure 4. The force is applied for 3 s and during this time Q travels a distance of 6 m. The coefficient of friction between each particle and the plane is . Find

the acceleration of Q,

the tension in the string.

State how in your calculation you have used the information that the string is inextensible.

When the particles have moved for 3 s, the force F is removed.

Find the time between the instant that the force is removed and the instant that Q comes to rest.

Two particles A and B, of mass m and 2 m respectively, are attached to the ends of a light inextensible string. The particle A lies on a rough horizontal table. The string passes over a small smooth pulley P fixed on the edge of the table. The particle B hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between A and the table is. The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of A and B is . By writing down separate equations of motion for A and B,

find the tension in the string immediately after the particles begin to move,

When B has fallen a distance h, it hits the ground and does not rebound. Particle A is then

Find the speed of A as it reaches P.

State how you have used the information that the string is light.