## Monday, December 12, 2016

### Rotational Dynamics Problems with Solutions Three

We are solving series of problems on rotational dynamics. Similar to equations of motion of translatory motion, we can write equations of rotational dynamics. Similar to mass we have moment of inertia and similar to force, we have torque in rotational dynamics. Similar to displacement we have angular displacement and similar to velocity we have angular velocity. Similar to conservation of angular momentum, we have conservation of angular momentum in rotational motion. If only centripetal force is acting in a motion then it is called uniform circular motion. If there is another acceleration called tangential acceleration, we call the body is in non uniform circular motion and it is studied under rotational dynamics.

Problem

A tangential force of 10 newton is acting on a circular plate of radius 50 centimeter such that it can rotate about an axis perpendicular to the plane passing through the center. If moment of inertia is given to us, we need to measure the number of rotations it makes in the first six seconds. Problem is as shown in the diagram below.

Solution

We can measure the torque acting on a system as the product of moment of inertia and the angular acceleration. Thus we can get the value of angular acceleration. Using that data, we can measure the angular displacement using the corresponding equation of motion as shown in the diagram below. Angular velocity can be written as the component of angular frequency and the problem can be solved as shown in the diagram below.

Problem

A ceiling fan is rotating about about its kw  axis with a uniform angular velocity. Electric current is switched off and due to constant opposing torque its angular velocity to reduced to a new value and it happens in thirty rotations. We need to measure the number of rotations that the body takes before it comes to state of rest.

Solution

We can find angular acceleration using the initial and final angular velocity and angular displacement. We can also use third equation of motion and further simplify the given data as shown in the diagram below.

Problem

A simple pendulum is oscillating with an angular amplitude 90 degree. If the direction of resultant acceleration of the bob is horizontal at a point whee angle made by the string with vertical is. Problem is as shown in the diagram below.

Solution

Body in circular motion shall be having centripetal acceleration and we know the velocity at a point making some angle as given in the problem. We also know that the sin component of acceleration due to gravity and we can find the tangential acceleration using the respective formula. By dividing both the cases, we can get the answer as shown in the diagram below.

Problem

A circular disc of known radius and thickness has a known moment of inertia and it is perpendicular to the plane and passing though its center. It is melted and recasted into a solid sphere. We need to find the moment of inertia and the problem is as shown in the diagram below.

Solution

Even when the body is recasted its volume remains same and it can be equated as shown in the diagram and hence we can find the relation between two radius of the systems. By applying the rules and formula of moment of inertia, we can solve the problem as shown in the diagram below. Moment of inertia is a property of a body that opposes the rotational motion of the body.

Problem

Four uniform rods each of mass M and length L are connected in the form of a square. We need to find the moment of inertia of the system about the geometrical axis.

Solution

Moment of inertia of rod can be expressed in terms of its length and mass and we can find the moment of inertia of the rod about the given axis using parallel axes theorem as shown in the diagram below. Moment of inertia of the system is four times of each rod as all of them are identical. The solution is as shown in the diagram below.

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