Motion down rough slope

5 A straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.

1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.

2 A basket of mass 5 kg slides down a slope inclined at \(12 ^ { \circ }\) to the horizontal. The coefficient of friction between the basket and the slope is 0.2 .

1. Find the frictional force acting on the basket.
2. Determine whether the speed of the basket is increasing or decreasing.

1 A straight ice track of length 50 m is inclined at \(14 ^ { \circ }\) to the horizontal. A man starts at the top of the track, on a sledge, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He travels on the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is 0.02 . Find the speed of the sledge and the man when they reach the bottom of the track.

2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .

1. Find the deceleration of the block.
2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.

3 A particle is released from rest and slides down a line of greatest slope of a rough plane which is inclined at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .

1. Find the acceleration of the particle.
2. Find the distance travelled by the particle in the first 3 s after it is released.

2. \begin{figure}[h] \end{figure} A rough plane is inclined at \(40 ^ { \circ }\) to the horizontal. Two points \(A\) and \(B\) are 3 metres apart and lie on a line of greatest slope of the inclined plane, with \(A\) above \(B\), as shown in Figure 2. A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the plane at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is released.

1. Find the acceleration of \(P\) down the plane.
2. Find the speed of \(P\) at \(B\).

5. \section*{Figure 3} A suitcase of mass 10 kg slides down a ramp which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The suitcase is modelled as a particle and the ramp as a rough plane. The top of the plane is \(A\). The bottom of the plane is \(C\) and \(A C\) is a line of greatest slope, as shown in Fig. 3. The point \(B\) is on \(A C\) with \(A B = 5 \mathrm {~m}\). The suitcase leaves \(A\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and passes \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the decleration of the suitcase,
2. the coefficient of friction between the suitcase and the ramp. The suitcase reaches the bottom of the ramp.
3. Find the greatest possible length of \(A C\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\)
• the coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \frac { 5 } { 12 }\)

The particle \(P\) is released from rest and slides down the plane.
Air resistance is modelled as being negligible.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on \(P\),
2. show that, as \(P\) slides down the plane, the acceleration of \(P\) down the plane is $$\frac { 1 } { 13 } g ( 5 - 12 \mu )$$
3. State what would happen to \(P\) if it is released from rest but \(\mu \geqslant \frac { 5 } { 12 }\)

12 A box of mass \(m \mathrm {~kg}\) slides down a rough slope inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the box and the slope is 0.4 . The box has an initial velocity of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slope. Calculate the distance the box travels before coming to rest.

3 A box, of mass 3 kg , is placed on a rough slope inclined at an angle of \(40 ^ { \circ }\) to the horizontal. It is released from rest and slides down the slope.

1. Draw a diagram to show the forces acting on the box.
2. Find the magnitude of the normal reaction force acting on the box.
3. The coefficient of friction between the box and the slope is 0.2 . Find the magnitude of the friction force acting on the box.
4. Find the acceleration of the box.
5. State an assumption that you have made about the forces acting on the box.

9. The diagram below shows a parcel, of mass \(m \mathrm {~kg}\), sliding down a rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 7 } { 25 }\). \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-24_394_906_497_584} The coefficient of friction between the parcel and the slope is \(\frac { 1 } { 12 }\). In addition to friction, the parcel experiences a variable resistive force of \(m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the parcel at time \(t\) seconds.

1. Show that the motion of the parcel satisfies the differential equation $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$

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The tile on a roof becomes loose and slides from rest down the roof. The roof is modelled as a plane surface inclined at 30° to the horizontal. The coefficient of friction between the tile and the roof is 0.4. The tile is modelled as a particle of mass \(m\) kg.

1. Find the acceleration of the tile as it slides down the roof. [7]

The tile moves a distance 3 m before reaching the edge of the roof.

1. Find the speed of the tile as it reaches the edge of the roof. [2]
2. Write down the answer to part (a) if the tile had mass \(2m\) kg. [1]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\includegraphics{figure_3} A small packet, of mass \(1.2\) kg, is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac{1}{8}\). When a force of magnitude \(8.4\) N, acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,

1. show that \(7(\cos \alpha + 8 \sin \alpha) = 40\). \hfill [6 marks]

Given that the solution of this equation is \(\alpha = 38°\),

1. find the acceleration with which the packet moves down the plane when it is released from rest with no external force applied. \hfill [4 marks]

A particle of mass \(m\) is placed on a rough inclined plane. The plane makes an angle \(\theta\) with the horizontal. The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.

1. Draw a fully labelled diagram to show the forces acting on the particle. [1]
2. Find an expression in terms of \(g\), \(\theta\) and \(\mu\) for the acceleration of the particle. [5]
3. Explain what would happen to the particle if \(\mu > \tan \theta\). [1]