Motion up then down slope

7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.

1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
3. Find the speed with which the particle returns to \(P\).

5 A particle \(P\) of mass 0.6 kg moves upwards along a line of greatest slope of a plane inclined at \(18 ^ { \circ }\) to the horizontal. The deceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the frictional and normal components of the force exerted on \(P\) by the plane. Hence find the coefficient of friction between \(P\) and the plane, correct to 2 significant figures. After \(P\) comes to instantaneous rest it starts to move down the plane with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(a\).

4. A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). A particle of mass 2 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is 0.25

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\). The particle now moves down the plane from \(A\).
3. Find the speed of \(P\) as it passes through \(O\).

1. A fixed rough plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)

A particle of mass 6 kg is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(B\).
2. Find the distance \(A B\). The particle now slides down the plane from \(B\). At the instant when the particle passes through the point \(C\) on the plane, the speed of the particle is again \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the distance \(B C\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-23_2647_1835_118_116}

7 A particle of mass 0.1 kg is at rest at a point \(A\) on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The particle is given an initial velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .

1. Find the coefficient of friction between the particle and the plane.
2. Show that, after coming to instantaneous rest, the particle moves down the plane.
3. Find the speed with which the particle passes through \(A\) during its downward motion.

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.

5. A small stone is projected with speed \(7 \mathrm {~ms} ^ { - 1 }\) from \(P\), the bottom of a rough plane inclined at \(25 ^ { \circ }\) to the horizontal, and moves up a line of greatest slope of the plane until it comes to instantaneous rest at \(Q\), where \(P Q = 4 \mathrm {~m}\).

1. Show that the deceleration of the stone as it moves up the plane has magnitude \(\frac { 49 } { 8 } \mathrm {~ms} ^ { - 2 }\).
2. Find the coefficient of friction between the stone and the plane,
3. Find the speed with which the stone returns to \(P\).
4. Name one force which you have ignored in your mathematical model, and state whether the answer to part (c) would be larger or smaller if that force were taken into account.

A particle of mass \(0.1\) kg is at rest at a point \(A\) on a rough plane inclined at \(15°\) to the horizontal. The particle is given an initial velocity of \(6\) m s\(^{-1}\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after \(1.5\) s.

1. Find the coefficient of friction between the particle and the plane. [7]
2. Show that, after coming to instantaneous rest, the particle moves down the plane. [2]
3. Find the speed with which the particle passes through \(A\) during its downward motion. [6]

A particle \(P\) of mass \(0.5\) kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is \(0.6\).

1. Show that the magnitude of the frictional force acting on \(P\) is \(2.25\) N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4\) m s\(^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]

A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6.

1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \text{ m s}^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]