Motion with friction on horizontal surface

8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.

1. Draw and label a diagram to show the forces acting on the crate.
2. Express the normal reaction between the surface and the crate in terms of \(T\).
3. Find \(T\).

A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v\,\text{m}\,\text{s}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).

1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{d}v}{\text{d}t} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]

A particle \(P\) of mass \(0.5\) kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t\) s the magnitude of the force is \(0.6t^2\) N and the velocity of \(P\) away from \(O\) is \(v \text{ ms}^{-1}\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).

1. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac{\text{dv}}{\text{dt}} = 1.2t^2 - 0.3 \quad \text{for } t > 0.5.$$ [3]
2. Express \(v\) in terms of \(t\) for \(t > 0.5\). [3]
3. Find the displacement of \(P\) from \(O\) when \(t = 1.2\). [3]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

\includegraphics{figure_3} A sledge has mass 30 kg. The sledge is pulled in a straight line along horizontal ground by means of a rope. The rope makes an angle \(20°\) with the horizontal, as shown in Figure 3. The coefficient of friction between the sledge and the ground is 0.2. The sledge is modelled as a particle and the rope as a light inextensible string. The tension in the rope is 150 N. Find, to 3 significant figures,

1. the normal reaction of the ground on the sledge, [3]
2. the acceleration of the sledge. [3]

When the sledge is moving at \(12 \text{ m s}^{-1}\), the rope is released from the sledge.

1. Find, to 3 significant figures, the distance travelled by the sledge from the moment when the rope is released to the moment when the sledge comes to rest. [6]

\includegraphics{figure_1} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude \(E\) newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N, and the acceleration of the truck is \(0.2 \text{ m s}^{-2}\). Find the value of \(E\). [4]

A go-kart and driver, of combined mass 55 kg, move forward in a straight line with a constant acceleration of \(0.2\text{ m s}^{-2}\) The total driving force is 14 N Find the total resistance force acting on the go-kart and driver. Circle your answer. [1 mark] 0N 3N 11N 14N

A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s\(^{-2}\), in a straight line, on a smooth horizontal surface. Find the weight of the crate. Circle your answer. [1 mark] 15 kg \quad 15g N \quad 15 N \quad 15g kg

A vehicle is driven at a constant speed of \(12\text{ ms}^{-1}\) along a straight horizontal road. Only one of the statements below is correct. Identify the correct statement. Tick (\(\checkmark\)) one box. The vehicle is accelerating The vehicle's driving force exceeds the total force resisting its motion The resultant force acting on the vehicle is zero The resultant force acting on the vehicle is dependent on its mass [1 mark]

Two constant forces act on a particle, of mass 2 kilograms, so that it moves forward in a straight line. The two forces are: • a forward driving force of 10 newtons • a resistance force of 4 newtons. Find the acceleration of the particle. Circle your answer. [1 mark] \(2 \text{ m s}^{-2}\) \(\quad\) \(3 \text{ m s}^{-2}\) \(\quad\) \(5 \text{ m s}^{-2}\) \(\quad\) \(12 \text{ m s}^{-2}\)