Chapter 3

A block of weight \(12 \mathrm{~N}\) rests in rough contact with a horizontal plane and \(\mu=1\). A force of \(3 \mathrm{~N}\) is applied horizontally to the block. The frictional force acting on the block is: (a) \(4 \mathrm{~N}\) (b) \(3 \mathrm{~N}\) (c) \(-4 \mathrm{~N}\) (d) zero because the block does not move.

Forces represented by \(2 \mathbf{i}+5 \mathbf{j}, \mathbf{i}-8 \mathbf{j}\) and \(p \mathbf{i}+q \mathbf{j}\) are in equilibrium, therefore: (a) \(p=3\) and \(q=-3\). (b) \(p=-3\) and \(q=3\) (c) \(p=-2\) and \(q=3\)

A small object of weight \(4 W\) in rough contact with a horizontal plane is acted upon by a force inclined at \(30^{\circ}\) to the plane. When the force is of magnitude \(2 W\) the object is about to slip. Calculate the magnitude of the normal reaction. and the coefficient of friction between the object and the plane.

Three forces \(\mathbf{F}_{1}, \mathbf{F}_{2}\) and \(\mathbf{F}_{3}\) are in equilibrium, therefore: (a) \(\mathbf{F}_{1}=\mathbf{F}_{2}+\mathbf{F}_{3}\) (b) \(\mathbf{F}_{1}+\mathbf{F}_{2}+\mathrm{F}_{3}=\mathbf{0}\) (c) \(F_{1}-F_{2}-F_{3}=0\) (d) \(-\mathbf{F}_{1}=\mathbf{F}_{2}+\mathbf{F}_{3}\)

A sphere of radius \(9 \mathrm{~cm}\) rests on a smooth inclined plane (angle \(30^{\circ}\) ). It is attached by a string fixed to a point on its surface to a point on the plane \(12 \mathrm{~cm}\) from the point of contact and on the same line of greatest slope. Find the tension. in the string if the weight of the sphere is \(100 \mathrm{~N}\).

Three concurrent forces represented by \(2 \mathrm{i}+3 \mathrm{j}, \mathrm{i}-4 \mathrm{j}\) and \(-3 \mathrm{i}+\mathrm{j}\) : (a) are in equilibrium, (b) have zero linear resultant, (c) have an equilibrant, (d) exert a tuming effect.

A weight \(W\) is suspended by two ropes which make \(30^{\circ}\) and \(60^{\circ}\) with the horizontal. If the tension in the first rope is \(20 \mathrm{~N}\), find the tension in the other and the value of \(W\).

A uniform rod of length \(l\) rests over the rim of a fixed hemispherical bowl of radius \(r\), with one end in contact with the surface of the bowl. If all contacts are smooth and the inclination of the rod to the horizontal is \(\theta\), prove that the value of \(\theta\) is given by the equation \(8 r \cos ^{2} \theta-l \cos \theta-4 r=0\).

A uniform rod \(A B\) of weight \(12 \mathrm{~N}\) is iree to tum in a vertical plane about a smooth hinge at its upper end \(\mathrm{A}\). It is held at an angle \(\theta\) to the vertical by a force \(P\) acting at \(\mathrm{B}\). (a) \(P\) is \(5 \mathrm{~N}\) applied horizon tally. What is the force at the hinge? (b) \(P\) is horizontal and \(\theta\) is arctan 2. What is the force at the hinge? (c) \(P\) is at right angles to \(\mathrm{AB}\) and of magnitude \(3 \mathrm{~N}\). What is the force at the hinge? (d) \(P\) is at right angles to \(\mathrm{AB}\) and \(\theta\) is aretan \(3 .\) Find \(P\) and the hinge force.

A cylinder of weight } 100 \mathrm{~N} \text { rests in the angle between a smooth vertical wall }and a smooth plane inclined at \(30^{\circ}\) to the wall. Find the thrusts of the cylinder on the wall and the plane.

A uniform rod \(A B\) of length \(l\) is in equilibrium at \(60^{\circ}\) to the horizontal, with its end \(\mathrm{A}\) on a horizontal plane, and resting against a fixed smooth sphere which has its lowest point C on the plane. The vertical plane containing the rod passes through the centre of the sphere. If the rod is on the point of slipping in the vertical plane, calculate the coefficient of friction between the rod and the plane when: (a) \(A C=1 l\) (b) \(\mathrm{AC}=5 l\)

A block rests on a rough inclined plane. Find the coefficient of friction between block and plane: (a) the weight of the block is \(8 \mathrm{~N}\), (b) the elevation of the plane is \(30^{\circ}\), (c) friction is limiting.

A uniform ladder of weight \(W\) rests on rough horizontal ground against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the ladder is inclined at an angle \(\alpha\) to the vertical. Prove that, if the ladder is on the point of slipping and \(\mu\) is the coefficient of friction between it and the ground, then \(\tan \alpha=2 \mu\).

If a frictional force acts on a body, it is not necessarily of value \(\mu R\) where \(R\) is the normal contact force.

A particle rests on a rough plane inclined at an angle \(\theta\) to the horizontal. The coefficient of friction between the particle and the plane is \(\mu\). When the weight of the particle is \(W\), a horizontal force of magnitude \(P\) just prevents the particle from slipping down the plane. If however a force of magnitude \(2 P\) acts parallel to the plane, the particle is on the point of slipping up the plane. The same force acting on a particle of weight \(2 W\) just prevents it from slipping down the same plane. Find the values of \(\theta\) and \(\mu\), and express \(P\) in terms of \(W\).