Circular Motion, Work, Power and Energy: NEET Physics Revision Notes

Circular Motion and Work, Power and Energy are important NEET Physics topics that test your understanding of forces, motion and energy transfer. Numerical questions can become challenging when you need to combine multiple concepts or identify the conditions that apply to a particular situation. Problems involving centripetal force, banking of roads, conical pendulums, vertical circles and spring compression often require careful analysis before you choose an equation.

These notes cover the key concepts, essential formulas and their applications to common NEET question types. Use the formula tables and problem-solving pointers to revise important relationships, understand their limitations and improve your approach to numerical problems involving circular motion and work, power and energy.

Circular Motion: Acceleration and Kinematics

Circular motion occurs when a particle moves along a circular path around a fixed centre. Its velocity changes continuously because its direction changes, even when its speed remains constant.

Tangential and Normal Acceleration

In curvilinear motion, acceleration can be resolved into two perpendicular components.

The total acceleration is the vector sum of these components. Since they are perpendicular,

[
a=\sqrt{a_t^2+a_n^2}
]

Remember:

• If tangential acceleration acts along the velocity, the particle speeds up.

• If it acts opposite to the velocity, the particle slows down.

• In uniform circular motion, tangential acceleration is zero, but normal acceleration is present.

Angular and Linear Quantities

Angular quantities describe rotational motion, while linear quantities describe the motion of a particle along its path.

Here, angles must be expressed in radians when using these relationships.

Equations of Angular Motion

For constant angular acceleration, the equations of rotational kinematics are analogous to the equations of linear motion.

Useful conversion: If rotational speed is given in revolutions per minute (RPM),

[\omega=\frac{2\pi N}{60}\ \text{rad/s}]

where (N) is the rotational speed in RPM.

Uniform vs Non-Uniform Circular Motion

In both cases, the direction of normal acceleration changes continuously as the particle moves around the circle.

Centripetal Force and Its Applications

A particle moving along a circular path requires a net inward force to provide centripetal acceleration. This force is called the centripetal force, but it is not a separate fundamental force. It is the resultant inward component of forces such as tension, gravity, friction or the normal reaction.

Centripetal Force Formulas

How to Solve Circular Motion Questions

Follow these steps when solving force-based numerical problems:

1. Identify the centre of the circular path.

2. Draw a free-body diagram showing all the forces.

3. Resolve forces along the radial and tangential directions.

4. Equate the net inward force to (mv^2/r).

5. Use energy conservation or tangential motion equations if the speed changes.

Common Applications of Centripetal Force

The equations in the table assume the stated force is the only contributor to the radial force balance. Always check the free-body diagram before applying them.

Conical Pendulum

A conical pendulum consists of a mass suspended by a string that moves in a horizontal circle. The string traces the surface of a cone.

For a string of length (L) making an angle (\theta) with the vertical:

Here, (T) represents string tension, while (T_p) represents the time period.

Particle Moving in a Bowl or Funnel

A particle moving along a smooth curved surface may travel in a horizontal circle. The normal reaction supplies the required centripetal force, while its vertical component balances the particle's weight.

For the angle (\theta) measured from the vertical: [ N\cos\theta=mg ]

 [ N\sin\theta=\frac{mv^2}{r}]

Dividing the equations gives: [ \tan\theta=\frac{v^2}{rg} ]

These relations apply when the particle remains in contact with the surface and the chosen geometry matches the angle definition.

Wall of Death: Object Inside a Rotating Cylinder

An object pressed against the inner wall of a rotating cylinder experiences a normal reaction towards the centre and friction along the wall.

The minimum angular speed is obtained when the available static friction is just sufficient to balance the weight.

Turning on Roads and Banking of Roads

When a vehicle turns, it requires an inward force to follow the curved path. On a level road, static friction provides this force. Banking allows a component of the normal reaction to contribute to the centripetal force.

Turning on a Level Road

For a vehicle of mass (m) travelling around a level circular road of radius (r):[ f_s=\frac{mv^2}{r}]

Since maximum static friction is (\mu_smg), the maximum safe speed is:

[v_{\max}=\sqrt{\mu_sgr}]

The maximum safe speed increases with the coefficient of friction and the square root of the turning radius.

Banking of Roads

A banked road is inclined at an angle (\theta) to the horizontal.

Here, (\mu_s) is the coefficient of static friction. The maximum and minimum speed formulas assume the usual banked-road model and valid contact conditions. If the expression for minimum speed becomes negative inside the square root, the vehicle can remain on the bank at zero speed under the idealised model.

Multiple Particles Connected by Strings

When several particles rotate together, each string provides the force needed for the masses beyond it.

Problem-solving approach:

• Draw a separate free-body diagram for each mass.

• Identify the radius of each particle's circular path.

• Apply the radial force equation to each mass.

• To find the tension in a string, consider the combined radial force requirement of the masses outside that string, accounting for all other radial forces.

This method avoids treating every string tension as equal when the system has multiple masses and different radii.

Work Done by a Force

Work measures the energy transferred by a force when an object undergoes displacement. The work done depends on the force, the displacement and the angle between them.

Work Done by Constant and Variable Forces

Positive, Negative and Zero Work

Important: The normal force and tension do not always do zero work. Their work is zero only when they are perpendicular to the displacement or when the relevant displacement is zero.

Work Done by Gravity and a Spring

For gravity, (h_i) and (h_f) are the initial and final heights. For a spring, (x_i) and (x_f) are the initial and final extensions or compressions measured from the natural length.

Gravity and ideal spring forces are conservative. Their work depends on the initial and final positions, not on the path taken.

Work-Energy Theorem and Potential Energy

The work-energy theorem connects the net work done on an object with the change in its kinetic energy.

Work-Energy Theorem

[W_{\text{net}}=\Delta K=\frac12mv_f^2-\frac12mv_i^2]

The theorem applies to the net work done by all forces. It can be used even when mechanical energy is not conserved.

Potential Energy

Potential energy is associated with the configuration of a system when conservative forces act.

For a conservative force, the change in potential energy is the negative of the work done by that force.

Conservative vs Non-Conservative Forces

When only conservative forces do work, the total mechanical energy remains constant:

[K_i+U_i=K_f+U_f]

When non-conservative forces such as friction do work, account for their work using the work-energy theorem or an energy balance.

Power: Average, Instantaneous and Constant Power

Power is the rate at which work is done or energy is transferred.

Important Power Formulas

If force acts perpendicular to velocity, its instantaneous power is zero. In non-uniform circular motion, the tangential component of the net force can change the speed and deliver power.

Motion Under Constant Power

For an object of constant mass moving in a straight line under a constant positive power, with the force in the direction of motion:

[P=Fv=mv\frac{dv}{dt}]

If it starts from rest, the following proportionalities apply:

These relationships assume constant power, constant mass, straight-line motion and an initial speed of zero. They should not be applied without checking these conditions.

Vertical Circular Motion: Tension, Speed and Energy

Vertical circular motion involves changes in both speed and the radial force required to maintain the path. Gravity helps the motion on the downward part of the circle and opposes it on the upward part. 

Consider a particle attached to a string moving in a vertical circle of radius (R). Let the bottom be the reference position for gravitational potential energy.

Forces at the Lowest and Highest Points

Here, (v_B) is the speed at the bottom, (v_T) is the speed at the top and (T) is the string tension.

Minimum Speed to Complete a Vertical Circle

For a particle just completing a vertical circle while attached to a string, the limiting condition at the top is (T=0).

These results follow from combining the radial force equation at the top with conservation of mechanical energy.

Energy Relation at Different Points

If the particle moves from the bottom to a point at height (h), conservation of mechanical energy gives: [\frac12mv_B^2=\frac12mv^2+mgh]

Therefore, [v^2=v_B^2-2gh] For the top of the circle, (h=2R), giving (v_T^2=v_B^2-4gR).

Important points:

• For just completing a circle with a string, the tension at the top is zero.

• If the speed is insufficient, the string may become slack before the particle completes the circle.

• On a rigid circular track, the normal reaction replaces tension, but the contact condition must still be checked.

• The minimum bottom speed (\sqrt{5gR}) applies to the idealised case of a particle moving on a track or attached to a string that remains taut as required.

Spring-Launched and Height-Based Circular Motion

Energy conservation can also be used when the particle enters circular motion after being launched by a spring or released from a height.

The height result assumes no energy loss and that the particle begins from rest. Use the work-energy theorem if other forces perform work.

Spring Compression Under a Falling Block

When a block of mass (m) is released from rest onto an initially uncompressed vertical spring, gravity does work while the spring compresses. At maximum compression, the block is momentarily at rest.

If the block begins at the point where it first touches the spring, the energy equation is:

[mgx=\frac12kx^2]

For non-zero compression, this gives:[x=\frac{2mg}{k}]

If the block falls through an additional height (h) before reaching maximum compression, use:

[mg(h+x)=\frac12kx^2]

Circular Motion and Work, Power and Energy become easier to approach when you identify the forces, displacement and energy changes involved in each problem. Revise the radial force equations, banking conditions, work-energy theorem, power relationships and vertical-circle speed requirements. Practise numerical questions using the formula tables, and check the assumptions and directions of forces before substituting values.