3.03v Motion on rough surface: including inclined planes

4. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).

1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.

3. \begin{figure}[h] \end{figure} A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(A B = 14 \mathrm {~m}\). At \(A\) the package has speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the package has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Find

1. the total energy lost by the package in travelling from \(A\) to \(B\),
2. the coefficient of friction between the package and the ramp.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is projected up a rough plane with initial speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point \(X\) on the plane, as shown in Figure 4. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 8 }\).

1. Use the work-energy principle to show that \(X Y = 25 \mathrm {~m}\). After reaching \(Y\), the particle \(P\) slides back down the plane.
2. Find the speed of \(P\) as it passes through \(X\).

A truck of mass 1800 kg is towing a trailer of mass 800 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 20 }\). The truck is connected to the trailer by a light inextensible rope which is parallel to the direction of motion of the truck. The resistances to motion of the truck and the trailer from non-gravitational forces are modelled as constant forces of magnitudes 300 N and 200 N respectively. The truck is moving at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the truck is working at a rate of 40 kW .

1. Find the value of \(v\). As the truck is moving up the road the rope breaks.
2. Find the acceleration of the truck immediately after the rope breaks.
   

8. The points \(A\) and \(B\) are 10 m apart on a line of greatest slope of a fixed rough inclined plane, with \(A\) above \(B\). The plane is inclined at \(25 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 5 kg is released from rest at \(A\) and slides down the slope. As \(P\) passes \(B\), it is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find, using the work-energy principle, the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the coefficient of friction between the particle and the plane.

1. A van of mass 900 kg is moving down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to motion of the van has constant magnitude 570 N . The engine of the van is working at a constant rate of 12.5 kW .

At the instant when the van is moving down the road at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 10 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) and \(A B = 6.5 \mathrm {~m}\), as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). The work done against friction as \(P\) moves from \(A\) to \(B\) is 245 J .

1. Find the value of \(\mu\). The particle is projected from \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle,
2. find the speed of the particle as it passes through \(B\).

A truck of mass 900 kg is towing a trailer of mass 150 kg up an inclined straight road with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The trailer is attached to the truck by a light inextensible towbar which is parallel to the road. The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 9 }\). The resistance to motion of the truck from non-gravitational forces has constant magnitude 200 N and the resistance to motion of the trailer from non-gravitational forces has constant magnitude 50 N .

1. Find the rate at which the engine of the truck is working. When the truck and trailer are moving up the road at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the towbar breaks, and the trailer is no longer attached to the truck. The rate at which the engine of the truck is working is unchanged. The resistance to motion of the truck from non-gravitational forces and the resistance to motion of the trailer from non-gravitational forces are still forces of constant magnitudes 200 N and 50 N respectively.
2. Find the acceleration of the truck at the instant after the towbar breaks.
3. Use the work-energy principle to find out how much further up the road the trailer travels before coming to instantaneous rest.

A truck of mass 750 kg is moving with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 49 }\). The resistance to motion of the truck is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .

1. Find the value of \(v\). On another occasion the truck is moving up the same straight road. The resistance to motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .
2. Find the acceleration of the truck at the instant when it is moving with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   

8. A particle \(P\) is projected up a line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is projected from the point \(O\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to instantaneous rest at the point \(A\). By Using the Work-Energy principle, or otherwise,

1. find, to 3 significant figures, the length \(O A\).
2. Show that \(P\) will slide back down the plane.
3. Find, to 3 significant figures, the speed of \(P\) when it returns to \(O\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 3 N . The other end of the string is attached to a fixed point \(O\) on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\) The coefficient of friction between \(P\) and the plane is \(\frac { \sqrt { 5 } } { 5 }\) The particle \(P\) is initially at rest at the point \(O\), as shown in Figure 8. The particle \(P\) then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle \(P\) moves up the plane and comes to rest at the point \(A\).

1. Find the extension of the elastic string when \(P\) is at \(A\).
2. Show that the particle does not remain at rest at \(A\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\) The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5. The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.

1. Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\). Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
2. find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
3. Describe the motion of \(P\) after it reaches \(B\), justifying your answer.

2. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)

1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
2. Find the acceleration of \(P\) when it is released from rest at \(B\).

6. Figure 2 \includegraphics[max width=\textwidth, alt={}, center]{c4b453e7-8a32-458b-8041-58c9e4ef9533-5_691_1067_241_584} A uniform solid cylinder has radius \(2 a\) and height \(\frac { 3 } { 2 } a\). A hemisphere of radius \(a\) is removed from the cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder, and the centre \(O\) of the hemisphere is also the centre of this plane face, as shown in Fig. 2. The remaining solid is \(S\).

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (6) The lower plane face of \(S\) rests in equilibrium on a desk lid which is inclined at an angle \(\theta\) to the horizontal. Assuming that the lid is sufficiently rough to prevent \(S\) from slipping, and that \(S\) is on the point of toppling when \(\theta = \alpha\),
2. find the value of \(\alpha\).
   (3) Given instead that the coefficient of friction between \(S\) and the lid is 0.8 , and that \(S\) is on the point of sliding down the lid when \(\theta = \beta\),
3. find the value of \(\beta\).
   (3)

1. A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m g\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\frac { 1 } { 4 } a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\frac { 1 } { 2 } a\).

By using the principle of conservation of energy,

1. find \(V\) in terms of \(a\) and \(g\),
2. find, in terms of \(a\) and \(g\), the speed of \(P\) when the string first becomes slack.
   

3. \begin{figure}[h] \end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\).

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(O A = \frac { 7 } { 6 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).

1. Show that \(\mu \geqslant \frac { 1 } { 2 }\). The particle is now moved along the table to the point \(B\), where \(O B = \frac { 3 } { 2 } l\), and released from rest. Given that \(\mu = \frac { 1 } { 2 }\), find
2. the speed of \(P\) at the instant when the string becomes slack,
3. the total distance moved by \(P\) before it comes to rest again.
   

A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 5 }\). The particle is held at rest at a point \(B\) on the plane, where \(O B = 1.5 \mathrm {~m}\). When \(P\) is at \(B\), the tension in the string is 20 N . The particle is released from rest.

1. Find the speed of \(P\) when \(O P = 1.2 \mathrm {~m}\). The particle comes to rest at the point \(C\).
2. Find the distance \(B C\).
   

4. \begin{figure}[h] \end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(3 m g\), is fixed to a point \(A\) on a fixed plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\) A small ball of mass \(2 m\) is attached to the free end of the string. The ball is held at a point \(C\) on the plane, where \(C\) is below \(A\) and \(A C = l\) as shown in Figure 3. The string is parallel to a line of greatest slope of the plane. The ball is released from rest. In an initial model the plane is assumed to be smooth.

1. Find the distance that the ball moves before first coming to instantaneous rest. In a refined model the plane is assumed to be rough. The coefficient of friction between the ball and the plane is \(\mu\). The ball first comes to instantaneous rest after moving a distance \(\frac { 2 } { 5 } l\).
2. Find the value of \(\mu\).

4. One end of a light elastic string, of modulus of elasticity \(2 m g\) and natural length \(l\), is fixed to a point \(O\) on a rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The other end of the string is attached to a particle \(P\) of mass \(m\) which is held at rest on the plane at the point \(O\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle is released from rest and slides down the plane, coming to instantaneous rest at the point \(A\), where \(O A = k l\). Given that \(k > 1\), find, to 3 significant figures, the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_152_72_118_127} \includegraphics[max width=\textwidth, alt={}]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_90_1620_123_203} □ ⟶ \(\_\_\_\_\) T

7. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The distance travelled down the chute by each brick is 8 m . A brick of mass 3 kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the potential energy lost by the brick in moving down the chute.
2. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
3. Hence find the coefficient of friction between the brick and the chute. Another brick of mass 3 kg slides down the chute. This brick is given an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the chute.
4. Find the speed of this brick when it reaches the bottom of the chute.

4 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_494_255_258_945} Particles \(P\) and \(Q\), of masses 0.4 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the sections of the string not in contact with the pulley are vertical. \(P\) rests in limiting equilibrium on a plane inclined at \(60 ^ { \circ }\) to the horizontal (see diagram).

1. (a) Calculate the components, parallel and perpendicular to the plane, of the contact force exerted by the plane on \(P\).
   (b) Find the coefficient of friction between \(P\) and the plane. \(P\) is held stationary and a particle of mass 0.2 kg is attached to \(Q\). With the string taut, \(P\) is released from rest.
2. Calculate the tension in the string and the acceleration of the particles. \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_579_1195_1553_475} The \(( t , v )\) diagram represents the motion of two cyclists \(A\) and \(B\) who are travelling along a horizontal straight road. At time \(t = 0 , A\), who cycles with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), overtakes \(B\) who has initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From time \(t = 0 B\) cycles with constant acceleration for 20 s . When \(t = 20\) her speed is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which she subsequently maintains.

7 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-4_129_798_756_676} A winch drags a \(\log\) of mass 600 kg up a slope inclined at \(10 ^ { \circ }\) to the horizontal by means of an inextensible cable of negligible mass parallel to the slope (see diagram). The coefficient of friction between the \(\log\) and the slope is 0.15 , and the \(\log\) is initially at rest at the foot of the slope. The acceleration of the \(\log\) is \(0.11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the tension in the cable. The cable suddenly breaks after dragging the log a distance of 10 m .
2. (a) Show that the deceleration of the log while continuing to move up the slope is \(3.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures.
   (b) Calculate the time taken, after the cable breaks, for the log to return to its original position at the foot of the slope. www.ocr.org.uk after the live examination series.
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7 A particle \(P\) of mass 0.6 kg is projected up a line of greatest slope of a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves with deceleration \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and comes to rest before reaching the top of the plane.

1. Calculate the frictional force acting on \(P\), and the coefficient of friction between \(P\) and the plane.
2. Find the magnitude of the contact force exerted on \(P\) by the plane and the angle between the contact force and the upward direction of the line of greatest slope,
   1. when \(P\) is in motion,
   2. when \(P\) is at rest.

6 A particle \(P\) of mass 0.3 kg is projected upwards along a line of greatest slope from the foot of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The initial speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction is 0.15 . The particle \(P\) comes to instantaneous rest before it reaches the top of the plane.

1. Calculate the distance \(P\) moves up the plane.
2. Find the time taken by \(P\) to return from its highest position on the plane to the foot of the plane.
3. Calculate the change in the momentum of \(P\) between the instant that \(P\) leaves the foot of the plane and the instant that \(P\) returns to the foot of the plane.