Newton’s laws of motion：練習問題

Two particles of masses \( 3 \text{ kg} \) and \( M \text{ kg} \) are connected by a light inextensible string passing over a smooth fixed pulley. The system is released from rest with the string taut. Given that the acceleration of the particles is \( 4 \text{ ms}^{-2} \) and \( M > 3 \), find the value of \( M \). (Take \( g = 10 \text{ ms}^{-2} \))

A particle of mass \( 2 \text{ kg} \) is on a rough plane inclined at \( 30^\circ \) to the horizontal. The coefficient of friction is \( 0.4 \). A horizontal force \( H \) acts on the particle. Find the minimum value of \( H \) that will prevent the particle from sliding down the plane. (Take \( g = 10 \text{ ms}^{-2} \))

A particle of mass \( 5 \text{ kg} \) is on a rough plane inclined at an angle \( \theta \) to the horizontal. When \( \theta = 25^\circ \), the particle is in limiting equilibrium and on the point of sliding down the plane. Calculate the coefficient of friction \( \mu \) between the particle and the plane.

A particle of mass \( 4 \text{ kg} \) is held on a smooth plane inclined at \( 30^\circ \) to the horizontal by a force \( H \) acting horizontally. If the particle is moving up the plane with an acceleration of \( 1.5 \text{ ms}^{-2} \), find the magnitude of \( H \). (Take \( g = 10 \text{ ms}^{-2} \))

A block of mass 5 kg is being pulled along a rough horizontal floor by a constant horizontal force of 22 N. Given that the coefficient of friction between the block and the floor is 0.3 and the acceleration of the block is constant, find this acceleration.

A block of mass \( 4 \text{ kg} \) is pulled along a rough horizontal floor by a horizontal force of \( 25 \text{ N} \). Given that the coefficient of friction between the block and the floor is \( 0.4 \) and taking \( g = 10 \text{ ms}^{-2} \), calculate the acceleration of the block.

Particle \( A \) of mass \( 2 \text{ kg} \) is on a smooth plane inclined at \( 30^\circ \) to the horizontal. It is connected by a light string over a smooth pulley at the top of the plane to particle \( B \) of mass \( 3 \text{ kg} \) hanging vertically. Find the acceleration of the system.

A crate of mass \( 20 \text{ kg} \) is being lowered vertically by a cable. The crate is decelerating at a constant rate of \( 2 \text{ ms}^{-2} \). Calculate the tension in the cable.

A particle of mass 4 kg is pulled along a rough horizontal floor by a constant force of magnitude 15 N. This force acts at an angle of \(30^\circ\) above the horizontal. The coefficient of friction between the particle and the floor is 0.25.
(a) Find the magnitude of the normal reaction force exerted by the floor on the particle.
(b) Calculate the acceleration of the particle.

A car of mass \( 1200 \text{ kg} \) is towing a caravan of mass \( 800 \text{ kg} \) down a straight road inclined at \( 5^\circ \) to the horizontal. The resistances to motion are \( 450 \text{ N} \) on the car and \( 300 \text{ N} \) on the caravan. The car's engine exerts a braking force (a force acting up the hill) of \( 800 \text{ N} \).
(a) Calculate the acceleration of the car and caravan.
(b) Find the magnitude of the force in the tow-bar and state whether it is a tension or a thrust.
(c) If the braking force is increased so that the system moves at a constant speed, find the new braking force.