Dynamics: Newton's laws, connected particles and friction

A puck P of mass 1.5 kg slides on a smooth horizontal surface. The only horizontal forces acting on P are (i + 6j) N and (−6i − 4j) N, where i and j are perpendicular horizontal unit vectors. P is modelled as a particle.

A stone of mass 0.8 kg is released from rest at the surface of a pond and sinks vertically. The water exerts a constant resistance of 2.4 N on the stone (this includes every force from the water). The stone is modelled as a particle. Take g = 9.8 m s⁻².

A goods lift of mass 500 kg carries a box of mass 30 kg on its floor. The lift is supported by a vertical cable. At one stage of the motion the tension in the cable is 4800 N. Take g = 9.8 m s⁻².

A small box slides in a straight line across a rough horizontal warehouse floor. It passes a point A with speed 6 m s⁻¹ and comes to rest 9 m beyond A. No force other than friction acts on the box horizontally. The box is modelled as a particle. Take g = 9.8 m s⁻².

A particle P of mass 0.5 kg lies on a smooth horizontal surface. Figure 1 shows the three horizontal forces F₁, F₂ and F₃ acting on P, drawn on a grid in which one square represents 1 N. The unit vectors i and j are horizontal and perpendicular, as shown.

A sledge of mass 40 kg is pulled from rest across rough horizontal snow by a horizontal rope with constant tension. After 6 s the rope is released and the sledge slides on until it comes to rest. Figure 1 is the velocity–time graph of the motion. The sledge is modelled as a particle and the coefficient of friction between the sledge and the snow is constant. Take g = 9.8 m s⁻².

A crate of mass 30 kg rests on a rough horizontal floor. A horizontal force of magnitude P newtons is applied to the crate and P is slowly increased from zero. Figure 1 shows how the magnitude F newtons of the friction force on the crate varies with P. The crate is modelled as a particle and, once it slides, the friction on it keeps its limiting value. Take g = 9.8 m s⁻².

A student of mass 60 kg stands on a set of bathroom scales on the floor of a lift. The scales show the magnitude of the normal reaction on the student. The lift starts from rest at time t = 0 and moves vertically upwards. Figure 1 shows the reading on the scales for 0 ≤ t ≤ 12, where t is in seconds. The student is modelled as a particle. Take g = 9.8 m s⁻².

A trunk of mass 30 kg is dragged across a rough horizontal floor by a rope. The rope is inclined at 20° above the horizontal and the tension in it is T newtons. The coefficient of friction between the trunk and the floor is 0.35. The trunk is modelled as a particle and the rope as light. Take g = 9.8 m s⁻².

A particle of mass 2.5 kg is on a smooth plane inclined at 25° to the horizontal. It is pushed up the plane by a horizontal force of magnitude 30 N, which acts in the vertical plane containing a line of greatest slope. The particle starts from rest. Take g = 9.8 m s⁻².

A car of mass 1350 kg tows a trailer of mass 450 kg up a straight road inclined at an angle α to the horizontal, where sin α = ¹⁄₁₅. The tow bar is light, rigid and parallel to the road. The resistances to motion are constant: 240 N on the car and 160 N on the trailer. The engine of the car produces a constant driving force of 3600 N. Take g = 9.8 m s⁻².

Two particles, of masses 0.6 kg and 0.4 kg, are attached to the ends of a light inextensible string that passes over a small smooth peg fixed above a horizontal floor. The particles hang with the hanging parts of the string vertical, the 0.6 kg particle being 0.9 m above the floor. The system is released from rest with the string taut. Take g = 9.8 m s⁻².

In a classroom demonstration, a block A of mass 5 kg lies on a rough horizontal bench. A light inextensible string attached to A passes over a small smooth pulley fixed at the edge of the bench and carries a block B of mass 3 kg, which hangs freely. The string between A and the pulley is horizontal and perpendicular to the edge of the bench. When the blocks are released from rest, a motion sensor shows that they accelerate at 2.1 m s⁻². The blocks are modelled as particles. Take g = 9.8 m s⁻².

A particle of mass 4 kg moves on a smooth horizontal plane under the action of three forces: F₁ = (pi + 3j) N, F₂ = (−2i + qj) N and F₃ = (5i − 7j) N, where p and q are constants. The particle starts from rest at time t = 0 and its acceleration is (1.5i − 0.5j) m s⁻².

Figure 1 shows particles A, B and C, of masses 2.2 kg, 1.5 kg and 1 kg. A and B are attached to the ends of a light inextensible string that passes over a smooth pulley fixed above them. C hangs below B, attached to it by a second light inextensible string. All hanging parts of the strings are vertical. The system is released from rest with both strings taut. No particle reaches the pulley or the ground during the motion described. Take g = 9.8 m s⁻².

A tractor pulls two trailers P and Q along a straight horizontal road. The tractor is joined to P, and P is joined to Q, by light rigid horizontal couplings. Figure 1 shows the masses of the three vehicles and the constant resistances to motion acting on them. The tractor's engine produces a constant driving force D newtons and the vehicles accelerate at 0.5 m s⁻².

A particle is projected up a line of greatest slope of a rough plane inclined at an angle α to the horizontal. The coefficient of friction between the particle and the plane is μ. Figure 1 is the velocity–time graph for the first 1.5 s of the motion, taking the direction up the plane as positive. Take g = 9.8 m s⁻².

A barge of mass 12 000 kg is pulled from rest along a straight canal by two horizontal ropes attached at its bow. Figure 1, a view from above, shows the forces on the barge: the ropes make angles of 30° and 40° with the centre line of the barge, on opposite sides, and a resistance of 2500 N acts along the centre line. The tension in the first rope is 4000 N and the tension in the second is T₂ newtons. The barge moves along its centre line. The barge is modelled as a particle.

Two boxes A and B, of masses 4 kg and 6 kg, are at rest in contact on a rough horizontal floor. The coefficient of friction between each box and the floor is 0.1. A horizontal force of magnitude P newtons is applied to A, pushing it against B as shown in Figure 1, which also shows how P varies with the time t seconds. The boxes are modelled as particles moving along the same straight line. Take g = 9.8 m s⁻².

Figure 1 shows a block of mass 4 kg on a rough plane inclined at 35° to the horizontal. The coefficient of friction between the block and the plane is 0.2. A horizontal force of magnitude 15 N acts on the block, towards the plane, in the vertical plane containing a line of greatest slope. The block is released from rest and slides down the plane. The block is modelled as a particle. Take g = 9.8 m s⁻².

A particle A of mass 3 kg is held at rest on a rough plane inclined at angle α to the horizontal, where tan α = ¾. The coefficient of friction between A and the plane is ¹⁄₆. A is attached to one end of a light inextensible string, which lies along a line of greatest slope of the plane and passes over a small smooth pulley fixed at the top of the plane. The other end of the string is attached to a particle B of mass 4 kg, which hangs freely 2 m above horizontal ground. A is released and the system moves with the string taut. A does not reach the pulley. Take g = 9.8 m s⁻².

A hospital lift of mass 800 kg carries one passenger of mass 65 kg. The lift starts from rest and moves vertically upwards. It accelerates uniformly for 3 s until its speed is 2.4 m s⁻¹, travels at 2.4 m s⁻¹ for 8 s, then decelerates uniformly, coming to rest after a further 2 s. The lift is raised by a vertical cable. Take g = 9.8 m s⁻².

A box of mass 12 kg is pulled up a line of greatest slope of a rough ramp inclined at 15° to the horizontal. The pulling rope is light and makes an angle of 25° with the ramp, in the vertical plane containing the line of greatest slope, above the ramp. The tension in the rope is 90 N and the coefficient of friction between the box and the ramp is 0.3. The box starts from rest. After 4 s the rope is let go. The box is modelled as a particle. Take g = 9.8 m s⁻².

A fixed wedge has two plane faces that meet at a horizontal ridge. Face X is smooth and inclined at 50° to the horizontal; face Y is rough and inclined at 30° to the horizontal. A small smooth pulley is fixed at the ridge. Particle P, of mass 5 kg, rests on face X and particle Q, of mass 2 kg, rests on face Y. They are joined by a light inextensible string that passes over the pulley, each part of the string lying along a line of greatest slope of its face. The coefficient of friction between Q and face Y is 0.25. The particles are released from rest with the string taut; P moves down face X and Q moves up face Y. Neither particle reaches the pulley or the bottom of its face during the motion described. Take g = 9.8 m s⁻².

Figure 1 shows two particles A and B on a rough plane inclined at 30° to the horizontal. A has mass 2 kg and the coefficient of friction between A and the plane is 0.3. B has mass 3 kg and the coefficient of friction between B and the plane is 0.1. A and B lie on the same line of greatest slope, with A higher than B, and are joined by a light rod parallel to the slope. The particles are released from rest and slide down the plane. Take g = 9.8 m s⁻².

A block A of mass 4 kg lies on a rough horizontal table. It is attached to one end of a light inextensible string that passes over a small smooth pulley fixed at the edge of the table. The other end of the string is attached to a particle B, which hangs freely at a height h metres above the floor, as shown in Figure 1. The string between A and the pulley is horizontal. The system is released from rest with the string taut. B hits the floor and does not rebound; A does not reach the pulley. Figure 1 also shows the velocity–time graph of A from release until A comes to rest. A and B are modelled as particles. Take g = 9.8 m s⁻².

A particle P of mass 2.5 kg moves on a smooth horizontal plane. The only horizontal force on P is a constant force F newtons. Figure 1 shows the velocity of P at time t = 0 and at time t = 4 s, drawn on a grid in which one square represents 1 m s⁻¹. The unit vectors i and j are horizontal and perpendicular, as shown.

A car tows a trailer down a straight road inclined at an angle α to the horizontal, where sin α = ¹⁄₂₀. The tow bar is light, rigid and parallel to the road. Figure 1 shows the masses of the car and the trailer and the constant resistances to motion acting on them. The car's engine provides no driving force, and a braking force of magnitude B newtons acts on the car only, up the road. The car and trailer are modelled as particles. Take g = 9.8 m s⁻².