Forces and equilibrium：練習問題

A block of mass 5 kg lies on a rough horizontal plane. The coefficient of friction between the block and the plane is \( \mu = 0.4 \). Taking \( g = 10 \text{ ms}^{-2} \), find the magnitude of the horizontal force required to just make the block begin to move.

A small box of mass \( 10\text{ kg} \) is held in equilibrium on a rough horizontal floor by a force of magnitude \( P \) acting at an angle of \( 30^\circ \) above the horizontal. The coefficient of friction between the box and the floor is \( 0.4 \). Given that the box is in limiting equilibrium and about to move, find the value of \( P \) using \( g = 10\text{ ms}^{-2} \).

A particle of mass 2 kg is placed on a rough plane inclined at \( 20^\circ \) to the horizontal. The coefficient of friction is \( 0.3 \). A force \( P \) acts up the plane parallel to the line of greatest slope. Find the maximum value of \( P \) for which the particle remains in equilibrium. (Use \( g = 10 \text{ ms}^{-2} \), \( \sin 20^\circ \approx 0.342 \), \( \cos 20^\circ \approx 0.940 \)).

A block of mass \( 12\text{ kg} \) rests in equilibrium on a smooth horizontal surface. A force of \( 50\text{ N} \) acts on the block at an angle of \( 20^\circ \) below the horizontal. Taking \( g = 10\text{ ms}^{-2} \), find the magnitude of the normal reaction force exerted by the surface on the block.

A particle of mass 4 kg is held in equilibrium on a smooth plane inclined at an angle of \( 30^\circ \) to the horizontal by a force of magnitude \( P \) N acting up the line of greatest slope. Taking \( g = 10 \text{ ms}^{-2} \), find the value of \( P \).

A force of \(25\text{ N}\) acts at an angle of \(60^{\circ}\) to the horizontal. Calculate the horizontal component of this force.

A block of mass \(8\text{ kg}\) is on a rough horizontal surface with coefficient of friction \(\mu = 0.25\). Calculate the magnitude of the horizontal force required to just start the block moving, taking \(g = 10\text{ ms}^{-2}\).

A particle of mass \(6\text{ kg}\) is in equilibrium on a smooth plane inclined at \(30^{\circ}\) to the horizontal, held by a force \(P\) acting parallel to the plane. Determine the magnitude of force \(P\) and the normal contact force \(R\) between the particle and the plane. (Take \(g = 10\text{ ms}^{-2}\))

A particle of mass \( 5 \text{ kg} \) is held in equilibrium on a smooth plane inclined at an angle \( \alpha \) to the horizontal by a force of magnitude \( 40 \text{ N} \) acting up the line of greatest slope. Taking \( g = 10 \text{ ms}^{-2} \):

(a) Find the value of \( \sin \alpha \).
(b) Find the magnitude of the normal reaction force exerted by the plane on the particle.

Two particles \( P \) and \( Q \), of masses \( 4 \text{ kg} \) and \( m \text{ kg} \) respectively, are connected by a light inextensible string. Particle \( P \) rests on a rough horizontal table where the coefficient of friction is \( 0.4 \). The string passes over a smooth pulley at the edge of the table, and particle \( Q \) hangs vertically. The system is initially in limiting equilibrium.

(a) Find the value of \( m \).
(b) An additional weight is added to \( Q \) so that its total mass becomes \( 5 \text{ kg} \). The system is released from rest. Find the acceleration of the system and the tension in the string during the motion. (Take \( g = 10 \text{ ms}^{-2} \))