Understanding Work Done in Circular Motion

The question asks about the work done on a body that is whirled in a circle. This scenario involves a fundamental concept in physics related to forces, displacement, and the definition of work.

What is Work Done?

In physics, work done by a force on an object is defined as the product of the force, the displacement of the object in the direction of the force, and the cosine of the angle between the force and the displacement vector. Mathematically, work (W) is given by:

\[ W = \vec{F} \cdot \vec{d} = Fd \cos\theta \]

where:

• \(F\) is the magnitude of the force,
• \(d\) is the magnitude of the displacement, and
• \(\theta\) is the angle between the force vector (\(\vec{F}\)) and the displacement vector (\(\vec{d}\)).

Forces in Circular Motion

When a body is whirled in a circle at a constant speed (uniform circular motion), it experiences a force that is always directed towards the center of the circle. This force is called the centripetal force. For example, if you swing a ball on a string, the tension in the string provides the centripetal force.

The key characteristic of the centripetal force in uniform circular motion is that it is always perpendicular to the instantaneous velocity of the object. Since displacement is in the direction of velocity for any infinitesimal movement, the centripetal force is also perpendicular to the instantaneous displacement.

Calculating Work Done in Circular Motion

Let's consider the work done by the centripetal force as the body moves along the circular path. At any point in the circular motion, the centripetal force (\(\vec{F}\)) is directed towards the center, along the radius, and the instantaneous displacement (\(\vec{d}\)) is along the tangent to the circle at that point, in the direction of the velocity.

The angle (\(\theta\)) between the centripetal force vector (\(\vec{F}\)) and the instantaneous displacement vector (\(\vec{d}\)) is always \(90^\circ\).

Now, let's use the work done formula:

\[ W = Fd \cos\theta \]

Substitute the angle \(\theta = 90^\circ\):

\[ W = Fd \cos(90^\circ) \]

We know that \(\cos(90^\circ) = 0\).

Therefore,

\[ W = Fd \times 0 \]

\[ W = 0 \]

The work done by the centripetal force on the body whirled in a circle is zero.

Implications of Zero Work Done

According to the work-energy theorem, the net work done on an object is equal to the change in its kinetic energy (\(\Delta KE\)).

\[ W_{net} = \Delta KE \]

If the only force doing work were the centripetal force, and its work done is zero, then the change in kinetic energy would be zero. This is consistent with uniform circular motion, where the speed (and thus kinetic energy, \(KE = \frac{1}{2}mv^2\)) remains constant.

It's important to note that while the centripetal force does no work, other forces might be acting (like tension, gravity if the circle is vertical, etc.). However, the question specifically asks about the work done on the body in general terms assuming the force maintaining circular motion is the primary focus for the work done calculation related to staying in a circle.

Summary of Work Done in Circular Motion

• Centripetal force is towards the center.
• Instantaneous displacement is tangential.
• The angle between force and displacement is \(90^\circ\).
• \(\cos(90^\circ) = 0\).
• Work done by the centripetal force is zero.

Therefore, if a body is whirled in a circle, the work done on it by the force causing the circular motion is zero.

Revision Table: Work, Force, and Motion

Additional Information: Work Done Examples

Understanding when work is done and when it is zero is crucial. Here are some examples:

• Positive Work: When force and displacement are in the same direction (\(\theta = 0^\circ\), \(\cos(0^\circ)=1\)). Example: Lifting a book upwards.
• Negative Work: When force and displacement are in opposite directions (\(\theta = 180^\circ\), \(\cos(180^\circ)=-1\)). Example: Work done by friction when a sliding object slows down.
• Zero Work: When force and displacement are perpendicular (\(\theta = 90^\circ\), \(\cos(90^\circ)=0\)). Example: Work done by gravity on a horizontally sliding object on a level surface, or the work done by the centripetal force in uniform circular motion as discussed here.
• Zero Work (Zero Displacement): If there is a force but no displacement (\(d=0\)), work done is zero. Example: Pushing against a stationary wall.

In the case of a body whirled in a circle by a centripetal force, the continuous \(90^\circ\) angle between the force and the instantaneous displacement ensures that the work done by this force is always zero.