Particle on inclined plane

1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the particle,
2. the value of \(\alpha\).

4 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438} \(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find

1. the values of \(u\) and \(a\),
2. the value of \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.

4 \includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_583_862_1343_644} Three coplanar forces of magnitudes \(68 \mathrm {~N} , 75 \mathrm {~N}\) and 100 N act at an origin \(O\), as shown in the diagram. The components of the three forces in the positive \(x\)-direction are \(- 60 \mathrm {~N} , 0 \mathrm {~N}\) and 96 N , respectively. Find

1. the components of the three forces in the positive \(y\)-direction,
2. the magnitude and direction of the resultant of the three forces. \(5 A , B\) and \(C\) are three points on a line of greatest slope of a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal, with \(A\) higher than \(B\) and \(B\) higher than \(C\). Between \(A\) and \(B\) the plane is smooth, and between \(B\) and \(C\) the plane is rough. A particle \(P\) is released from rest on the plane at \(A\) and slides down the line \(A B C\). At time 0.8 s after leaving \(A\), the particle passes through \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A particle is projected from a point \(P\) with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope \(P Q R\) of a rough inclined plane. The distances \(P Q\) and \(Q R\) are both equal to 0.8 m . The particle takes 0.6 s to travel from \(P\) to \(Q\) and 1 s to travel from \(Q\) to \(R\).

1. Show that the deceleration of the particle is \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and hence find \(u\), giving your answer as an exact fraction.
2. Given that the plane is inclined at \(3 ^ { \circ }\) to the horizontal, find the value of the coefficient of friction between the particle and the plane.

4 A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\). The particle travels 2 m during the first 0.8 s after passing through \(P\), then a further 6 m in the next 1.2 s . Find

1. the value of \(u\) and the acceleration of the particle,
2. the value of \(\alpha\) in degrees.

7 A horizontal force of 24 N acts on a block of mass 12 kg on a horizontal plane. The block is initially at rest. This situation is first modelled assuming the plane is smooth.

1. Write down the acceleration of the block according to this model. The situation is now modelled assuming a constant resistance to motion of 15 N .
2. Calculate the acceleration of the block according to this new model. How much less distance does the new model predict that the block will travel in the first 4 seconds? The 24 N force is removed and the block slides down a slope at \(5 ^ { \circ }\) to the horizontal. The speed of the block at the top of the slope is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 7. The answers to parts (iii) and (iv) should be found using the assumption that the resistance to the motion of the block is still a constant 15 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-5_255_901_1128_575} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
3. Calculate the acceleration of the block in the direction of its motion.
4. For how much time does the block slide down the slope before coming to rest and how far does it slide in that time? Measurements show that the block actually comes to rest in 3.5 seconds.
5. Assuming that the error in the prediction is due only to the value of the resistance, calculate the true value of the resistance.

8 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h] \end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The resistance to motion remains 100 N .
2. Find the speed of the sledge
   (A) 1.6 seconds after the rope breaks,
   (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_259_853_1457_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.

6 A cyclist freewheels, with a constant acceleration, in a straight line down a slope. As the cyclist moves 50 metres, his speed increases from \(4 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~ms} ^ { - 1 }\).

1. 1. Find the acceleration of the cyclist.
   2. Find the time that it takes the cyclist to travel this distance.
2. The cyclist has a mass of 70 kg . Calculate the magnitude of the resultant force acting on the cyclist.
3. The slope is inclined at an angle \(\alpha\) to the horizontal.
   1. Find \(\alpha\) if it is assumed that there is no resistance force acting on the cyclist.
   2. Find \(\alpha\) if it is assumed that there is a constant resistance force of magnitude 30 newtons acting on the cyclist.
4. Make a criticism of the assumption described in part (c)(ii).

3 A toy car is placed at the top of a ramp. After the car has been released from rest, it travels a distance of 1.08 metres down the ramp, in a time of 1.2 seconds. Assume that there is no resistance to the motion of the car.

1. Find the magnitude of the acceleration of the car while it is moving down the ramp.
2. Find the speed of the car, when it has travelled 1.08 metres down the ramp.
3. Find the angle between the ramp and the horizontal, giving your answer to the nearest degree.
   [0pt] [4 marks]

8 A crane lifts a crate of mass 20 kg using a light inextensible cable. The crate starts from rest and ascends 10 metres in 4 seconds during which time a constant tension of \(T \mathrm {~N}\) is applied in the cable. Find the value of \(T\).