3.03r Friction: concept and vector form

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

2 A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When P has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on P , find

1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the coefficient of friction between the particle and the plane.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assuming the plane is smooth,

1. find the speed of \(P\) at \(B\). The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
2. find the coefficient of friction between \(P\) and the plane.

4. \begin{figure}[h] \end{figure} A uniform ladder, of weight \(W\) and length \(2 a\), rests in equilibrium with one end \(A\) on a smooth horizontal floor and the other end \(B\) on a rough vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the ladder is \(\mu\). The ladder makes an angle \(\theta\) with the floor, where \(\tan \theta = 2\). A horizontal light inextensible string \(C D\) is attached to the ladder at the point \(C\), where \(A C = \frac { 1 } { 2 } a\). The string is attached to the wall at the point \(D\), with \(B D\) vertical, as shown in Fig. 2. The tension in the string is \(\frac { 1 } { 4 } W\). By modelling the ladder as a rod,

1. find the magnitude of the force of the floor on the ladder,
2. show that \(\mu \geqslant \frac { 1 } { 2 }\).
3. State how you have used the modelling assumption that the ladder is a rod.

4. \begin{figure}[h] \end{figure} Figure 3 shows a thin hollow right circular cone fixed with its circular rim horizontal.
The centre of the circular rim is \(O\). The vertex \(V\) of the cone is vertically below \(O\).
The radius of the circular rim is \(4 a\) and \(O V = 3 a\).
A particle \(P\) of mass \(m\) moves in a horizontal circle of radius \(r ( 0 < r < 4 a )\) on the inner surface of the cone. The coefficient of friction between \(P\) and the inner surface of the cone is \(\frac { 1 } { 4 }\) The particle moves with a constant angular speed.
Show that the maximum possible angular speed is \(\sqrt { \frac { 16 g } { 13 r } }\)

1. A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point \(A\) on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 7 }\)

The particle \(P\) is projected from \(A\), with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface of the table.
After travelling 0.8 m from \(A\), the particle passes through the point \(B\) on the table.

1. Find the speed of \(P\) at the instant it passes through \(B\). The particle \(P\) comes to rest at the point \(C\) on the table, where \(A B C\) is a straight line.
2. Find the total distance travelled by \(P\) as it moves directly from \(A\) to \(C\).
3. Show that \(P\) remains at rest at \(C\).

1. \begin{figure}[h] \end{figure} A rough disc is rotating in a horizontal plane with constant angular speed 20 revolutions per minute about a fixed vertical axis through its centre \(O\). A particle \(P\) rests on the disc at a distance 0.4 m from \(O\), as shown in Figure 1. The coefficient of friction between \(P\) and the disc is \(\mu\). The particle \(P\) is on the point of slipping. Find the value of \(\mu\).

5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:

1. \(\theta = 0\);
2. \(\theta = 20\);
3. \(\theta = 70\);
4. \(\theta = 90\).

8 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_502_935_1411_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

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8 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_305_1191_1050_701} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_355_1294_2156_429} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the trolleys, stating whether this rod is in tension or thrust (compression).

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

4 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422} The diagram shows a central cross-section CDEF of a uniform solid cube of weight \(W\) and with edges of length \(2 a\). The cube rests on a rough horizontal plane. A thin uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(6 a\), is hinged to the plane at \(A\). The rod rests in smooth contact with the cube at \(C\), with angle \(C A D\) equal to \(30 ^ { \circ }\). The rod is in the same vertical plane as \(C D E F\). The coefficient of friction between the plane and the cube is \(\mu\). Given that the system is in equilibrium, show that \(\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3\). [6] Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

4 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-3_567_575_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),

1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).

5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.

1. Find, in terms of \(d , g\) and \(\theta\),
   1. the magnitude of the vertical component of the force exerted on the rod by the ground,
   2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
2. find the value of \(d\).
   

3. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley, \(P\), fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(B\) hangs freely below \(P\), as shown in Figure 1. The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\) The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.

1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
3. Suggest two refinements to the model that would make it more realistic.

2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.

1. \begin{figure}[h] \end{figure} Figure 1 shows a particle \(P\) of mass 0.5 kg at rest on a rough horizontal plane.

1. Find the magnitude of the normal reaction of the plane on \(P\). The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 7 }\) A horizontal force of magnitude \(X\) newtons is applied to \(P\).
   Given that \(P\) is now in limiting equilibrium,
2. find the value of \(X\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\)
• the coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \frac { 5 } { 12 }\)

The particle \(P\) is released from rest and slides down the plane.
Air resistance is modelled as being negligible.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on \(P\),
2. show that, as \(P\) slides down the plane, the acceleration of \(P\) down the plane is $$\frac { 1 } { 13 } g ( 5 - 12 \mu )$$
3. State what would happen to \(P\) if it is released from rest but \(\mu \geqslant \frac { 5 } { 12 }\)

3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
The particle is being pulled by a horizontal force of 10 N .
It remains at rest in equilibrium. \begin{figure}[h] \end{figure}

1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.

11 A block of mass 2 kg is placed on a rough horizontal table. A light inextensible string attached to the block passes over a smooth pulley attached to the edge of the table. The other end of the string is attached to a sphere of mass 0.8 kg which hangs freely. The part of the string between the block and the pulley is horizontal. The coefficient of friction between the table and the block is 0.35 . The system is released from rest.

1. Draw a force diagram showing all the forces on the block and the sphere.
2. Write down the equations of motion for the block and the sphere.
3. Show that the acceleration of the system is \(0.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the time for the block to slide the first 0.5 m . Assume the block does not reach the pulley.

12 A box of mass \(m \mathrm {~kg}\) slides down a rough slope inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the box and the slope is 0.4 . The box has an initial velocity of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slope. Calculate the distance the box travels before coming to rest.