Uniform circular motion and gravitation：練習問題

A particle moves in a circular path with constant speed. Which of the following quantities remains constant?

The gravitational force between two point masses is \( F \) when they are separated by a distance \( d \). If the mass of one object is doubled and the distance between the two objects is also doubled, what is the new gravitational force between them?

A small block of mass $$m$$ is attached to a light rigid rod of length $$L$$. The block is given just enough speed at the bottom of a vertical circle to complete the loop. What is the magnitude and direction of the force exerted by the rod on the block at the highest point of the circle? (Neglect air resistance and assume $$g$$ is the acceleration due to gravity)

A particle performs uniform circular motion with a constant speed \( v \) in a circular path of radius \( r \). Which of the following expressions correctly represents the period \( T \) of the motion?

Planet X has a mass four times that of Earth and a radius twice that of Earth. What is the gravitational field strength on the surface of Planet X compared to Earth's gravitational field strength $$g_E$$?

Define the term gravitational field strength \( g \) at a point in space.

Calculate the gravitational field strength at a distance of \( 3R_E \) from the center of the Earth, where \( R_E \) is the Earth's radius and the field strength at the surface is \( 9.81 \text{ m s}^{-2} \).

A small bead slides without friction along a smooth wire bent into a vertical circle of radius \( R \). If the bead is released from the top of the circle, calculate its speed when it reaches the bottom in terms of \( g \) and \( R \).

A small toy car of mass \(0.40\text{ kg}\) travels along a track in a theme park model. It moves at a constant speed of \(1.8\text{ m s}^{-1}\) on a horizontal circular curve of radius \(0.90\text{ m}\).

(a) Calculate the magnitude of the centripetal acceleration of the toy car. (2 marks)

(b) State the origin of the centripetal force acting on the toy car. (1 mark)

(c) Determine the magnitude of the centripetal force required to keep the car moving along the curve. (1 mark)

A cylindrical amusement park ride, often called a 'rotor', has a radius of \(3.0\text{ m}\). A rider of mass \(60\text{ kg}\) stands against the inner wall of the cylinder. As the cylinder rotates about its vertical axis, the floor is lowered, but the rider remains pinned against the wall without sliding down. The coefficient of static friction between the rider's clothing and the wall is \(0.40\). Assume the acceleration due to gravity is \(g = 9.81\text{ m s}^{-2}\).

(a) Identify the horizontal force acting on the rider that provides the necessary centripetal acceleration. State its direction. (1 point)

(b) Explain, with reference to the forces acting on the rider, how the rotation of the cylinder prevents the rider from sliding downwards after the floor is removed. (2 points)

(c) Calculate the minimum angular velocity \(\omega\) (in \(\text{rad s}^{-1}\)) at which the cylinder must rotate to prevent the rider from sliding. (2 points)