Motion with applied force on slope

7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.

1. Given that the plane is smooth, find
   1. the acceleration of the particle,
   2. the change in kinetic energy after the particle has moved 12 m up the plane.
   3. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
      (a) Find the acceleration of the particle.
      (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held at rest on the plane by the action of a force of magnitude 4 N acting up the plane in a direction parallel to a line of greatest slope of the plane, as shown in Figure 2. The particle is on the point of slipping up the plane.

1. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 4 N is removed.
2. Find the acceleration of \(P\) down the plane.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N , acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.

1. Show that \(\cos \theta = \frac { 3 } { 5 }\).
2. Find the normal reaction between \(P\) and the plane. The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.
3. Find the initial acceleration of \(P\). \(\_\_\_\_\)}

3 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-2_300_501_799_790} A particle \(P\) of mass 0.25 kg moves upwards with constant speed \(u \mathrm {~ms} ^ { - 1 }\) along a line of greatest slope on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The pulling force acting on \(P\) has magnitude \(T \mathrm {~N}\) and acts at an angle of \(20 ^ { \circ }\) to the line of greatest slope (see diagram). Calculate

1. the value of \(T\),
2. the magnitude of the contact force exerted on \(P\) by the plane. The pulling force \(T \mathrm {~N}\) acting on \(P\) is suddenly removed, and \(P\) comes to instantaneous rest 0.4 s later.
3. Calculate \(u\).

4 A block is being pulled up a rough plane inclined at an angle of \(22 ^ { \circ }\) to the horizontal by a rope parallel to the plane, as shown in the diagram. The mass of the block is 0.7 kg , and the tension in the rope is \(T\) newtons. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-3_264_460_1649_779}

1. Draw a diagram to show the forces acting on the block.
2. Show that the normal reaction force between the block and the plane has magnitude 6.36 newtons, correct to three significant figures.
3. The coefficient of friction between the block and the plane is 0.25 . Find the magnitude of the frictional force acting on the block during its motion.
4. The tension in the rope is 5.6 newtons. Find the acceleration of the block.

1. \begin{figure}[h] \end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point \(O\) which lies on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\) as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from \(O\) down the line of greatest slope of the plane.

1. Find the tension in the spring.
2. Find the initial acceleration of the particle.

A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20°\) to the horizontal.

1. Show that the friction force acting on the particle is 0.684 N, correct to 3 significant figures. [1]

The coefficient of friction between the particle and the plane is 0.6. A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.

1. Find this acceleration. [4]

\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

1. Find the value of \(P\). [8]

The tension in the rope is now increased to 150 N.

1. Find the acceleration of the box. [6]

Lizzie is sat securely on a wooden sledge. The combined mass of Lizzie and the sledge is \(M\) kilograms. The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle \(\theta\) above the horizontal and remains taut as the sledge moves forward. \includegraphics{figure_17} The sledge remains in contact with the surface throughout. The coefficient of friction between the sledge and the surface is \(\mu\) and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, \(a \text{ m s}^{-2}\) The tension in the rope is a constant \(T\) Newtons.

1. Show that $$T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$$ [7 marks]
2. It is known that when \(M = 30\), \(\theta = 30°\), and \(T = 40\), the sledge remains at rest. Lizzie uses these values with the relationship formed in part (a) to find the value for \(\mu\) Explain why her value for \(\mu\) may be incorrect. [2 marks]

An object of mass 60 kg is on a rough plane inclined at an angle of 20° to the horizontal. The coefficient of friction between the object and the plane is \(0 \cdot 3\). Initially, the object is held at rest. A force which is parallel to the plane and of magnitude \(T\) N is applied to the object in an upward direction along the line of greatest slope. The object is then released.

1. Given that \(T = 15\), calculate the acceleration of the object down the plane. [6]
2. Given that \(T = 350\), determine whether or not the object moves up the plane. Give a reason for your answer. [3]