Circular Motion: Types, Centripetal Force, Angular Motion & Applications

TAKEAWAYS:

• Circular motion is motion along a curved path where the speed may be constant but velocity changes direction.
• Centripetal force is the net force required to keep an object moving in a circular path, always directed toward the center.
• Angular velocity (ω) measures how fast an object rotates, while linear velocity (v) is the speed along the path.
• Real-world examples include car tires, planets orbiting the sun, and spinning tops.
• Centrifugal force is a fictitious force that appears in rotating reference frames (e.g., a spinning merry-go-round).
• Equations like $v=r\omega$, $a_c=\frac{v^2}{r}$, and $F_c=m\frac{v^2}{r}$ are essential for solving problems.

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What is Circular Motion?

Circular motion is the movement of an object along a circular path. It can be uniform (constant speed) or non-uniform (changing speed). Even if the speed is constant, the velocity changes because its direction keeps changing.

Key Terms:

• Radius (r): Distance from the center to the object.
• Linear velocity (v): Speed along the path (m/s).
• Angular velocity (ω): How fast the object rotates (rad/s).
• Centripetal acceleration (a_c): Acceleration toward the center (m/s²).
• Centripetal force (F_c): Force required to keep the object moving in a circle (N).

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Angular Velocity (ω)

Angular velocity measures how fast an object rotates. It is related to linear velocity by: $$v=r\omega$$ where:

• $v$ = linear velocity (m/s),
• $r$ = radius (m),
• $\omega$ = angular velocity (rad/s).

Example 1: Calculating Angular Velocity

A car wheel of radius 0.4 m rotates at 100 m/s. Find its angular velocity.

Solution: $$\omega =\frac{v}{r}=\frac{100}{0.4}=250\text{ rad/s}$$

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Centripetal Acceleration (a_c)

Even if speed is constant, the object accelerates because its direction changes. The centripetal acceleration is: $$a_c=\frac{v^2}{r}=r{\omega }^2$$

Example 2: Calculating Centripetal Acceleration

A stone tied to a string of length 1.5 m is whirled at 20 m/s. Find its centripetal acceleration.

Solution: $$a_c=\frac{v^2}{r}=\frac{20^2}{1.5}=\frac{400}{1.5}=266.67{\text{ m/s}}^2$$

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Centripetal Force (F_c)

Centripetal force is the net force required to keep an object moving in a circle. It is given by: $$F_c=m\frac{v^2}{r}=mr{\omega }^2$$ where $m$ = mass of the object (kg).

Example 3: Calculating Centripetal Force

A 2 kg ball is tied to a string and rotated at 5 m/s in a circle of radius 1 m. Find the centripetal force.

Solution: $$F_c=m\frac{v^2}{r}=2\times \frac{5^2}{1}=2\times 25=50\text{ N}$$

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Centrifugal Force (Fictitious Force)

Centrifugal force is a pseudo-force that appears to act outward when observed from a rotating frame of reference (e.g., a spinning merry-go-round). It is not a real force but arises due to inertia.

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Applications of Circular Motion

1. Planetary Motion: Planets orbit the sun due to gravitational centripetal force.
2. Car Safety: Friction provides centripetal force to keep cars moving in curves.
3. Centrifuges: Used in labs to separate substances based on density.
4. Spinning Tops: A spinning top stays upright due to centripetal force.

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Solving Problems: Step-by-Step

1. Identify given quantities (v, r, ω, m, etc.).
2. Choose the correct formula (e.g., $v=r\omega$, $F_c=m\frac{v^2}{r}$).
3. Substitute values and solve.
4. Check units (m/s, rad/s, N, etc.).

Example 4: NEB-Style Problem

A stone of mass 0.5 kg is tied to a string and rotated in a vertical circle of radius 1 m at 4 m/s. Find the centripetal force.

Solution: $$F_c=m\frac{v^2}{r}=0.5\times \frac{4^2}{1}=0.5\times 16=8\text{ N}$$

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Common Mistakes to Avoid

• Confusing centripetal and centrifugal force: Centrifugal force is fictitious.
• Ignoring units: Always check if answers are in m/s, rad/s, or N.
• Assuming uniform circular motion: If speed changes, use calculus (not covered here).

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NEB Board-Style Questions

Short Answer (5 marks)

1. Define angular velocity. How is it related to linear velocity?
2. Why does a car skid when taking a sharp turn at high speed?
3. Explain the difference between centripetal and centrifugal force.

Long Answer (10 marks)

4. A 1 kg ball is rotated in a circle of radius 2 m at 3 rad/s.
   • Find its linear velocity.
   • Calculate the centripetal force acting on it.
   • If the string breaks, in which direction will the ball move?

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Exam Tip

• Memorize formulas: $v=r\omega$, $a_c=\frac{v^2}{r}$, $F_c=m\frac{v^2}{r}$.
• Draw diagrams: Always sketch the circular path and label forces.
• Practice numericals: NEB often tests calculations, so solve past papers.
• Understand concepts: Know why centripetal force is needed and why centrifugal force is fictitious.

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Gravitational force acts as centripetal force for planetary motion. (Image: Johnny Silvercloud, CC BY-SA 2.0, via Wikimedia Commons)