Motion in a circle：練習問題

A point on a spinning disc is located \(0.040 \text{ m}\) from the centre. If the disc rotates at a constant angular speed of \(30 \text{ rad s}^{-1}\), what is the linear speed of this point?

A ball of mass \(0.5\, kg\) is whirled in a horizontal circle of radius \(0.8\, m\) by a string. If the ball makes \(2.0\) revolutions per second, what is the tension in the string?

A mass is attached to a string and swings in a horizontal circle, forming a conical pendulum. The length of the string is \(L\), and the angle the string makes with the vertical is \(\theta\). What is the angular speed of the mass?

Which statement correctly defines the radian?

A cooling fan blade completes \(1200\) revolutions every minute. What is the angular speed of the blade in \(\text{rad s}^{-1}\)?

State the relationship between the linear speed \(v\) and the angular speed \(\omega\) of a particle moving in a circle of radius \(r\), and name the SI unit for angular displacement.

A particle of mass \(0.50 \, \text{kg}\) moves in a horizontal circle of radius \(0.40 \, \text{m}\) at a constant linear speed of \(2.0 \, \text{m s}^{-1}\). Calculate the centripetal force acting on the particle.

A pilot of mass m performs a vertical loop-the-loop of radius r at a constant speed v. Determine the ratio of the normal contact force at the bottom point P to the normal contact force at the top point Q of the loop.

(a) Define the radian.
(b) A small ball is attached to a string and whirled in a horizontal circle of radius \(0.65 \text{ m}\) with a constant angular speed. The ball completes \(20\) full revolutions in \(15 \text{ s}\). Calculate the angular speed \(\omega\) of the ball in \(\text{rad s}^{-1}\).
(c) Determine the linear speed \(v\) of the ball.

A ball of mass \(0.25 \text{ kg}\) is attached to a string of length \(1.2 \text{ m}\). The ball describes a horizontal circle with the string making an angle of \(30^\circ\) to the vertical.

Assume the acceleration of free fall \(g = 9.81 \text{ m s}^{-2}\).

(a) Draw a free-body diagram for the ball, showing all forces acting on it.

(b) Calculate the tension in the string.

(c) Determine the linear speed of the ball.