Circular motion：練習問題

A particle moves in a horizontal circle of radius \(2.0 \text{ m}\) at a constant speed. Given that the magnitude of the centripetal acceleration of the particle is \(8.0 \text{ m/s}^2\), calculate the speed of the particle.

A particle moves in a horizontal circle on the inside of a smooth hollow sphere of radius \(R\). The plane of the circle is at a depth \(\frac{1}{2}R\) below the center of the sphere. Find the speed of the particle.

A particle moves in a horizontal circle on the inside surface of a smooth hemispherical bowl of radius \(R\). The bowl is fixed with its rim horizontal and its vertex downwards. The particle moves in a circle at a height \(h\) above the vertex. Find the constant angular speed \(\omega\) of the particle in terms of \(g, R,\) and \(h\).

A particle moves along a circular path of radius \(0.8 \text{ m}\) with a constant speed of \(4 \text{ m/s}\). What is the magnitude of the centripetal acceleration of the particle?

A particle of mass \(m\) is attached to one end of a light inextensible string of length \(L\). The other end is fixed at point \(A\). The particle is released from rest when the string is taut and makes an angle of \(60^\circ\) with the downward vertical. Find the tension in the string when the particle passes through the lowest point of its path.

A particle of mass \(0.5\text{ kg}\) is attached to the end of a light inextensible string of length \(1.2\text{ m}\). The particle is rotating in a horizontal circle on a smooth horizontal table with the other end of the string fixed. If the tension in the string is \(6\text{ N}\), find the angular speed of the particle.

A bead of mass \(0.2 \text{ kg}\) is threaded on a smooth circular wire of radius \(0.4 \text{ m}\) fixed in a vertical plane. Given that the speed of the bead at the lowest point of the wire is \(4 \text{ m s}^{-1}\), find the magnitude of the normal contact force exerted by the wire on the bead at this point. [Take \(g = 10 \text{ m s}^{-2}\)]

A particle of mass \(m\) is attached to one end of a light rigid rod of length \(L\), which is free to rotate in a vertical plane about its other end. Find the minimum speed \(u\) at which the particle must be projected from the lowest point to just reach the highest point.

A car of mass \(m\) travels around a circular track of radius \(R = 50 \text{ m}\) which is banked at an angle of \(20^\circ\) to the horizontal.

Find the speed \(v\) at which the car can travel around the track such that there is no sideways frictional force between the tires and the track surface.

A small bead of mass \(m\) is threaded on a smooth circular wire of radius \(r\) fixed in a vertical plane. The bead is projected from the lowest point of the wire with speed \(\sqrt{3gr}\).

Find the magnitude of the normal contact force exerted by the wire on the bead when the bead has reached a height \(r\) above the center of the wire.