3.03t Coefficient of friction: F <= mu*R model

2 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_195_719_1114_712} A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T \mathrm {~N}\) (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25 , find the value of \(T\).
[0pt] [5]

2 A particle of mass 0.1 kg is released from rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. It is given that, 5 seconds after release, the particle has a speed of \(2 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the particle and hence show that the magnitude of the frictional force acting on the particle is 0.302 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

1 A particle of mass 2 kg is initially at rest on a rough horizontal plane. A force of magnitude 10 N is applied to the particle at \(15 ^ { \circ }\) above the horizontal. It is given that 10 s after the force is applied, the particle has a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the magnitude of the frictional force is 8.96 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

5 A particle of mass \(m \mathrm {~kg}\) is resting on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude 10 N applied to the particle up a line of greatest slope of the plane is just sufficient to stop the particle sliding down the plane. When a force of 75 N is applied to the particle up a line of greatest slope of the plane, the particle is on the point of sliding up the plane. Find \(m\) and the coefficient of friction between the particle and the plane.

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.

1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.

4 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. \(P\) is released from rest at a point on the table 3.5 m from \(O\). The speed of \(P\) at the instant the string becomes slack is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the work done against friction during the period from the release of \(P\) until the string becomes slack,
2. the coefficient of friction between \(P\) and the table.

2 A uniform solid cone has height 38 cm .

1. Write down the distance of the centre of mass of the cone from its base. \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840} The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted, and the cone remains in equilibrium until the angle of inclination of the plane reaches \(31 ^ { \circ }\) (see diagram), when the cone topples.
2. Find the radius of the cone.
3. Show that \(\mu \geqslant 0.601\), correct to 3 significant figures, where \(\mu\) is the coefficient of friction between the cone and the plane.

7 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).

1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
   1. the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
   2. the speeds of the particles.

2 A horizontal turntable rotates with constant angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle of mass 0.06 kg is placed on the turntable at a point 0.25 m from its centre. The coefficient of friction between the particle and the turntable is \(\mu\). As the turntable rotates, the particle moves with the turntable and no sliding takes place.

1. Find the vertical and horizontal components of the contact force exerted on the particle by the turntable.
2. Show that \(\mu \geqslant 0.225\).

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\),
2. the least possible value of the coefficient of friction at \(B\).

6 A horizontal disc with a rough surface rotates about a fixed vertical axis which passes through the centre of the disc. A particle \(P\) of mass 0.2 kg is in contact with the surface and rotates with the disc, without slipping, at a distance 0.5 m from the axis. The greatest speed of \(P\) for which this motion is possible is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the coefficient of friction between the disc and \(P\). \(P\) is now attached to one end of a light elastic string, which is connected at its other end to a point on the vertical axis above the disc. The tension in the string is equal to half the weight of \(P\). The disc rotates with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) rotates with the disc without slipping. \(P\) moves in a circle of radius 0.5 m , and the taut string makes an angle of \(30 ^ { \circ }\) with the horizontal.
2. Find the greatest and least values of \(\omega\) for which this motion is possible.
3. Calculate the value of \(\omega\) for which the disc exerts no frictional force on \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-3_656_757_781_694} The diagram shows the cross-section \(A B C D E F\) through the centre of mass of a uniform prism which rests with \(A B\) on rough horizontal ground. \(A B C D\) is a rectangle with \(A B = C D = 0.4 \mathrm {~m}\) and \(B C = A D = 1.8 \mathrm {~m}\). The other part of the cross-section is a semicircle with diameter \(D F\) and radius \(r \mathrm {~m}\).

1. Given that the prism is on the point of toppling, show that \(r = 0.6\). A force of magnitude \(P \mathrm {~N}\) is applied to the prism, acting at \(60 ^ { \circ }\) to the upwards vertical along a tangent to the semicircle at a point between \(D\) and \(E\). The prism has weight 15 N and is in equilibrium on the point of toppling about \(B\).
2. Show that \(P = 3.26\), correct to 3 significant figures.
3. Find the smallest possible value of the coefficient of friction between the prism and the ground.

1 \includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-02_560_421_258_861} A hollow cylinder with a rough inner surface has radius 0.5 m . A particle \(P\) of mass 0.4 kg is in contact with the inner surface of the cylinder. The particle and cylinder rotate together with angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about the vertical axis of the cylinder, so that the particle moves in a horizontal circle (see diagram). Given that \(P\) is about to slip downwards, find the coefficient of friction between \(P\) and the surface of the cylinder.

6. \begin{figure}[h] \end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.

1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
2. Find the greatest possible value of \(P\).
3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).

7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.

1. Find the magnitude of the acceleration of \(B\) immediately after release.
2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.

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 small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.

1. Draw a diagram showing all the forces acting on the parcel.
2. Find the normal reaction on the parcel.
3. Find the coefficient of friction between the parcel and the plane.

4. \begin{figure}[h] \end{figure} A parcel of mass 5 kg lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The parcel is held in equilibrium by the action of a horizontal force of magnitude 20 N , as shown in Fig. 2. The force acts in a vertical plane through a line of greatest slope of the plane. The parcel is on the point of sliding down the plane. Find the coefficient of friction between the parcel and the plane.
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6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the frictional force acting on \(P\) as it moves up the plane,
2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous 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.

5. \begin{figure}[h] \end{figure} A small ring of mass 0.25 kg is threaded on a fixed rough horizontal rod. The ring is pulled upwards by a light string which makes an angle \(40 ^ { \circ }\) with the horizontal, as shown in Figure 3. The string and the rod are in the same vertical plane. The tension in the string is 1.2 N and the coefficient of friction between the ring and the rod is \(\mu\). Given that the ring is in limiting equilibrium, find

1. the normal reaction between the ring and the rod,
2. the value of \(\mu\).

7. \begin{figure}[h] \end{figure} A package of mass 4 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the package,
2. the coefficient of friction between the plane and the package.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find

1. the acceleration of \(Q\),
2. the value of \(\mu\),
3. the tension in the string.
4. State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.

3. \begin{figure}[h] \end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the box,
2. the coefficient of friction between the box and the plane.