Question: A constant net torque is exerted on an object. Which of the following quantities for the object cannot be constant? Choose all that apply. (a) angular position (b) angular velocity (c) angular acceleration (d) moment of inertia (e) kinetic energy
The quantities that cannot be constant if a constant net torque is exerted on an object are:
Torque is defined as a measure of force that can cause an object to rotate about an axis. Torque is what causes an object to acquire angular acceleration. It is a vector quantity.
The magnitude of torque ‘ ’produced by a force’ F’is given by
‘ r’ is the length of the moment arm
‘ ’ is the angle between the force vector and the moment arm.
Angular velocity: This will change at a rate equal to the torque.
Angular position: If the angular velocity changes, the angular position will also change.
Kinetic energy: When a torque is applied, the linear or angular speed changes at a rate proportional to the torque. So the kinetic energy will also change.
Many machines employ cams for various purposes, such as opening and closing valves. In Figure P10.46, the cam is a circular disk of radius R with a hole of diameter R cut through it. As shown in the figure, the hole does not pass through the center of the disk. The cam with the hole cut out has mass M. The cam is mounted on a uniform, solid, cylindrical shaft of diameter R and also of mass M. What is the kinetic energy of the cam–shaft combination when it is rotating with angular speed v about the shaft’s axis?
A spool of thread consists of a cylinder of radius with end caps of radius as depicted in the end view shown in Figure P10.91. The mass of the spool, including the thread, is , and its moment of inertia about an axis through its center is l. The spool is placed on a rough, horizontal surface so that it rolls without slipping when a force acting to the right is applied to the free end of the thread. (a) Show that the magnitude of the friction force exerted by the surface on the spool is given by
(b) Determine the direction of the force of friction.
Big Ben, the nickname for the clock in Elizabeth Tower (named after the Queen in 2012) in London, has an hour hand long with a mass of 60.0 kg and a minute hand 4.50 m long with a mass of 100 kg (Fig. P10.49). Calculate the total rotational kinetic energy of the two hands about the axis of rotation. (You may model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.)
80. A common demonstration, illustrated in Figure P10.80, consists of a ball resting at one end of a uniform board of length , that is hinged at the other end and elevated at an angle. A light cup is attached to the board at. so that it will catch the ball when the support stick is removed suddenly. (a) Show that the ball will lag behind the falling board when is less than.
(b) Assuming the board is long and is supported at this limiting angle, show that the cup must be from the moving end.
To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance . Show that the radius of a given portion of the track is given by
where is the radius of the innermost portion of the track and is the angle through which the disc turns to arrive at the location of the track of radius . (b) Show that the rate of change of the angle is given by
where is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
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