A wheel of radius is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is . A mass less cord wrapped around the wheel is attached to a block that slides on a horizontal frictionless surface. If a horizontal force of magnitude is applied to the block as shown in Fig. , what is the magnitude of the angular acceleration of the wheel? Assume the cord does not slip on the wheel.
The magnitude of the angular acceleration of the wheel is
Due to applied force , the block will accelerate. So, using Newton’s second law, write a net force equation for the block. Similarly, use Newton’s second law for rotating the wheel. Then by using the relation between angular acceleration and linear acceleration, find the magnitude of the angular acceleration of the wheel.
Formulae are as follow:
where, T, P are forces, m is mass, R is radius, I is moment of inertia, a is acceleration and is angular acceleration.
Taking rightward motion to be positive for the block and clockwise motion to be negative for the wheel.
Applying Newton’s second law to the block gives,
Similarly, applying Newton’s second law to the wheel gives,
As tangential acceleration is opposite to that of block’s acceleration ,
Use equation (2) into (1),
Hence, the magnitude of the angular acceleration of the wheel is
Therefore, the magnitude of the angular acceleration of the wheel can be found using Newton’s second law for the block and for the rotating wheel by using the relation between angular acceleration and linear acceleration.
In Fig., two blocks, of mass and , are connected by a massless cord that is wrapped around a uniform disk of mass and radius . The disk can rotate without friction about a fixed horizontal axis through its centre; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks, (b) the tension in the cord at the left, and (c) the tension in the cord at the right.
A rigid body is made of three identical thin rods, each with length, fastened together in the form of a letter H (Fig.). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical?
A disk, with a radius of , is to be rotated like a merrygo-round through , starting from rest, gaining angular speed at the constant rate through the first and then losing angular speed at the constant rate until it is again at rest. The magnitude of the centripetal acceleration of any portion of the disk is not to exceed .
(a) What is the least time required for the rotation?
(b) What is the corresponding value of ?
Figure shows an early method of measuring the speed of light that makes use of a rotating slotted wheel. A beam of light passes through one of the slots at the outside edge of the wheel, travels to a distant mirror, and returns to the wheel just in time to pass through the next slot in the wheel. One such slotted wheel has a radius of and slots around its edge. Measurements taken when the mirror is from the wheel indicate a speed of light of . (a) What is the (constant) angular speed of the wheel? (b) What is the linear speed of a point on the edge of the wheel?
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