Kinetic Energy of a Rigid body
This note provides us an information about kinetic energy of rigid body in two cases kinetic energy in case of pure rotation and kinetic energy of a rolling object along a plane surface.
Summary
This note provides us an information about kinetic energy of rigid body in two cases kinetic energy in case of pure rotation and kinetic energy of a rolling object along a plane surface.
Things to Remember
- units of K.E. is ergs in cgs system joules in MKS SYSTEM and foot pounds in FPS system
- kinetic energy of a rolling object along a plane surface. $$=\frac{MV^{2}}{2}(\frac{R^{2}+K^{2}}{R^{2}})$$
- kinetic energy of rigid body in two cases kinetic energy in case of pure rotation \begin{align*} (T)=\frac{I\omega ^{2}}{2}\end{align*}
MCQs
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Subjective Questions
Q1:
What do you mean by delegation of authority?
Type: Short Difficulty: Easy
<p>According to <strong>Louis A. Allen,</strong>" Delegation is the dynamics of management, it is the process a manager follows in dividing the work assigned to him so that he performs that part which only he because of his unique organizational placement, can perform effectively and so that he can get others to help him with what remains."</p>
<p>According to <strong>Douglas C. Baril</strong>, "Delegation refers to a manager's ability to share his burden with others. It consists of granting authority or the right to decision making in certain defined areas and charging the subordinates with responsibility for carrying through an assigned task.'</p>
<p>In conclusion, the delegation of authority is the means of granting authority and responsibility regarding the specific job to the subordinates for the achievement organizational goals. It is based on the principles of division of work. It is an essential tool for effective organization.</p>
Q2:
Describe the principle of delegation of authority.
Type: Long Difficulty: Easy
<li><strong>Principle of parity of authority and responsibility: </strong>It is the most important principle of delegation of authority. Authority and responsibility should always go hand in hand. If a subordinate is given authority to do a particular task. he should be equally responsible for doing the task in a proper manner. If the authority is more than there will be misuse of authority and if there is more responsible than there will be burden of work. </li>
</ul>
<p> </p>
<ul>
<li><strong>Principle of absoluteness of responsibility: </strong>The term ‘responsibility’ means an obligation to do the jobs or works. According to this principle, authority can be delegated but responsibility cannot be delegated by a manager. The manager is responsible for the acts of his subordinate to whom he delegates authority for any work.</li>
</ul>
<p> </p>
<ul>
<li><strong>Unity of command:</strong> According to this principle, there should not be more heads commanding one head. In other words, there should be the single boss or superior in an organization. It means a subordinate should be under the command of only one superior at a time. One superior should delegate authority to only on subordinate at a time.</li>
</ul>
<p> </p>
<ul>
<li><strong>Principle of a functional definition of authority and responsibility:</strong> Authority and responsibility should be well-defined so that, to be clear about the limit of one’s rights, duties, responsibility, and accountability. If it could be, it will help the subordinates to learn the limits of one's rights, duties, responsibility, and accountability.</li>
</ul>
<p> </p>
<ul>
<li><strong>The scalar chain:</strong> According to this principle, authority flow from top to down. This scalar chain is the basis of the relationship between senior and subordinated. There should be unbroken chain of auhtority from top to bottom in the organizational structure.</li>
</ul>
<p> </p>
<ul>
<li><strong>Management by exception:</strong> Management by exception is a way for managers to effective save time and more efficiently run their department or business. Management by exception usually is most effective when managers have control over the problem areas. That way they can change processes to improve the company. </li>
</ul>
<p> </p>
<ul>
<li><strong>Principle of delegation by result expected:</strong> The nature and extent f authority delegated to a subordinate should be consistent with the results expected of him. The manager must claer about the results expected from the subordinates as per his delegation of authority.</li>
</ul>
<p> </p>
<ul>
<li><strong>Performance monitoring : </strong>The performance of the subordinate should be monitored by the superior because to make them move in a right direction towards achieving organizational goal. It also makes delegation of authority effective.</li>
</ul>
Q3:
What are the major barriers of effective delegation of authority? Explain.
Type: Long Difficulty: Easy
<li><strong>Reluctance to delegate: </strong>Manager may feel the reluctance to delegate authority. Some managers are disorganized that they are unable to plan work in advance and as a result authority cannot delegate properly.</li>
</ul>
<p> </p>
<ul>
<li><strong>Fear of subordinates: </strong>Some managers may fear that their subordinate may do well and threaten to their own advancement. They have low self-confidence and fear competition. Due to this, they don't like to delegate the authority.</li>
</ul>
<p> </p>
<ul>
<li><strong>Lack of trust: </strong>Some managers have conservative attitude and traditional faith. They do not believe their subordinate. They believe in “Do it yourself” principle. If they feel subordinates are not capable of accepting the responsibility, they may not take chance. Thus, they may not delegate the authority.</li>
</ul>
<p> </p>
<ul>
<li><strong>Incompetence of subordinates: </strong>Some employee may not want to accept delegated authority because of lack of self-confidence. They fear of making mistake in their performance. A manager may feel their subordinate have limited knowledge and ability to do their jobs or works. As a result, managers or superior do not want to delegate rights to their subordinates.</li>
</ul>
<p> </p>
<ul>
<li><strong>Distorted delegation:</strong> Managers may assign full responsibility but delegate inadequate authority to discharge these responsibilities. In such situation, it is difficult to perform the task.</li>
</ul>
<p> </p>
<ul>
<li><strong>Improper reward system: </strong>Delegation of authority means taking additional responsibility. If the organization has improper reward system then the subordinate do not want to take additional responsibility. As a result<strong>, </strong>delegation of authority may be obstructed.</li>
</ul>
<p> </p>
<ul>
<li><strong>Lack of control</strong><strong>: </strong>Some managers may think that they will lose control by delegating authority to their subordinates. They feel that if they delegate the authority to their subordinate, they would not be sure to achieve assigned responsibility from subordinate.</li>
</ul>
<p> </p>
<p> </p>
<p> </p>
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Kinetic Energy of a Rigid body
Kinetic Energy of a Rigid body:
a) Kinetic energy in case of pure rotation:
Consider a rigid body rotating about an axis with the angular velocity ω. Let the axis of rotation passes through the CG of the body. In case of pure rotation CG of the body doesn’t have any linear velocity and hence the body has only angular velocity.
Suppose that the rigid body is composed of large no. of elementary particles having masses m1, m2, m3…….mn at the distances r1, r2, r3………………..rn respectively from the axis of rotation. Here by the definition of rigid body, each particle has the same angular velocity equal to that of the rigid body. But the linear velocity is different for different particles due to their positions. i.e.
\begin{align*}v_{1}=r_{1}\omega ,v_{2}=r_{2}\omega ,v_{3}=r_{3}\omega ,..........,v_{n}=r_{n}\omega\end{align*}
So that, total kinetic energy of the body is given by,
\begin{align*}T=\frac{m_{1}v_{1}^{2}}{2}+\frac{m_{2}v_{2}^{2}}{2}+.......+\frac{m_{n}v_{n}^{2}}{2}\end{align*}\begin{align*}T=\frac{\omega ^{2}}{2}(m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+......+m_{n}r_{n}^{2})\end{align*}\begin{align*}=\frac{\omega ^{2}}{2}\sum m_{i}r_{i}^{2}\end{align*}\begin{align*}\therefore (T)=\frac{I\omega ^{2}}{2}\end{align*}
units of K.E. is ergs in cgs system joules in MKS SYSTEM and foot pounds in FPS system
b) Kinetic energy of a rolling object along a plane surface:
Let us consider a rigid body rolling on a plane surface. Thus the objects possess both the linear and angular velocity at the same time. Thus, kinetic energy of this body is the sum of kinetic energy due to both the linear and angular velocity. i.e.
K.E (T) = KE due to linear velocity + KE due to angular velocity.
$$=\frac{MV^{2}}{2}+\frac{I\omega ^{2}}{2}$$
Substituting the value of I and R we get
$$I= MK^2$$ and
$$\omega = \frac VR$$
$$=\frac{MV^{2}}{2}(\frac{R^{2}+K^{2}}{R^{2}})$$
Where, M= mass of the rolling object.
V= linear velocity of the object.
ω= angular velocity of the object.
I= M.I of the rolling object (considered spherical having radius R)
K= Radius of gyration.
References:
Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.
Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.
Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.
Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014
Lesson
Dynamics of Rigid Bodies
Subject
Physics
Grade
Bachelor of Science
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