Coefficient of friction from motion

3 A block of mass 8 kg slides down a rough plane inclined at \(30 ^ { \circ }\) to the horizontal, starting from rest. The coefficient of friction between the block and the plane is \(\mu\). The block accelerates uniformly down the plane at \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Draw a diagram showing the forces acting on the block.
2. Find the value of \(\mu\).
3. Find the speed of the block after it has moved 3 m down the plane.

2 A particle of mass 0.1 kg is released from rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. It is given that, 5 seconds after release, the particle has a speed of \(2 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the particle and hence show that the magnitude of the frictional force acting on the particle is 0.302 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

6 A block, of mass 5 kg , slides down a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. When modelling the motion of the block, assume that there is no air resistance acting on it.

1. Draw and label a diagram to show the forces acting on the block.
2. Show that the magnitude of the normal reaction force acting on the block is 37.5 N , correct to three significant figures.
3. Given that the acceleration of the block is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the coefficient of friction between the block and the plane.
4. In reality, air resistance does act on the block. State how this would change your value for the coefficient of friction and explain why.

\(A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30°\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(AB\) is \(2.25 \text{ m}\). A particle \(P\), of mass \(m \text{ kg}\), is released from rest at \(A\) and reaches \(B\) \(1.5 \text{ s}\) later. Find the coefficient of friction between \(P\) and the plane. [6]

A small box of mass 5 kg is pulled at a constant speed of \(2.5 \text{ m s}^{-1}\) down a line of greatest slope of a rough plane inclined at \(10°\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

1. Find the coefficient of friction between the box and the plane. [5]

The pulling force is removed while the box is moving at \(2.5 \text{ m s}^{-1}\).

1. Find the distance moved by the box after the instant at which the pulling force is removed. [4]

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

1. the acceleration of the particle, [3]
2. the coefficient of friction between the particle and the plane. [5]

The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}

1. Find the value of \(X\). [7]

\includegraphics{figure_3} A particle \(P\) of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points \(A\) and \(B\), where \(AB = 10\) m, as shown in Figure 3. The speed of \(P\) at \(A\) is 2 m s\(^{-1}\). The particle \(P\) takes 3.5 s to move from \(A\) to \(B\). Find

1. the speed of \(P\) at \(B\), [3]
2. the acceleration of \(P\), [2]
3. the coefficient of friction between \(P\) and the plane. [5]

\includegraphics{figure_4} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30°\) and \(60°\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.

1. Find the acceleration of \(P\) down the slope. [3 marks]
2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. [5 marks] \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \text{ m s}^{-2}\),
3. find, correct to 3 significant figures, the value of \(\mu\). [5 marks] In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
4. Find the value of \(t\) correct to 2 decimal places. [4 marks]