Unit 2.6 – Uniform Circular Motion

• Definition: Uniform circular motion is the motion of an object along a circular path at a constant speed.
• Key Characteristics:
  • The magnitude of velocity (speed) remains constant.
  • The direction of velocity changes continuously, always tangential to the circular path.
  • Acceleration is directed towards the center of the circle (centripetal acceleration).

2. Key Terms in Uniform Circular Motion

(a) Angular Displacement ($𝜃$):

• Definition: The angle swept by the radius vector as the object moves along the circular path.
• Formula:
  $$𝜃=\frac{\text{Arc Length (s)}}{\text{Radius (r)}}$$
• SI Unit: Radian (rad).

(b) Angular Velocity ($𝜔$):

• Definition: The rate of change of angular displacement with respect to time.
• Formula:
  $$𝜔=\frac{\mathrm{\Delta }𝜃}{\mathrm{\Delta }𝑡}$$
• SI Unit: Radian per second ($\text{rad/s}$).

(c) Linear Velocity ($𝑣$):

• Definition: The speed of the object moving along the circular path.
• Formula:
  $$𝑣=𝑟\cdot 𝜔$$
• Direction: Always tangential to the circular path.

(d) Centripetal Acceleration (${𝑎}_{𝑐}$):

• Definition: The acceleration directed towards the center of the circular path, responsible for changing the direction of velocity.
• Formula:
  $${𝑎}_{𝑐}=\frac{{𝑣}^2}{𝑟}=𝑟\cdot {𝜔}^2$$
• Direction: Always towards the center of the circle.

(e) Centripetal Force (${𝐹}_{𝑐}$):

• Definition: The net force acting towards the center of the circular path, required to keep the object in uniform circular motion.
• Formula:
  $${𝐹}_{𝑐}=𝑚\cdot {𝑎}_{𝑐}=\frac{𝑚\cdot {𝑣}^2}{𝑟}=𝑚\cdot 𝑟\cdot {𝜔}^2$$
• Direction: Always towards the center of the circle.

3. Differences Between Linear Motion and Circular Motion

Quick Revision Points

• Uniform Circular Motion: Constant speed, changing velocity direction.
• Angular Velocity: $𝜔=\frac{\mathrm{\Delta }𝜃}{\mathrm{\Delta }𝑡}$.
• Linear Velocity: $𝑣=𝑟\cdot 𝜔$.
• Centripetal Acceleration: ${𝑎}_{𝑐}=\frac{{𝑣}^2}{𝑟}=𝑟\cdot {𝜔}^2$.
• Centripetal Force: ${𝐹}_{𝑐}=\frac{𝑚\cdot {𝑣}^2}{𝑟}=𝑚\cdot 𝑟\cdot {𝜔}^2$.

Previous Year Questions and Answers

• Radius ($𝑟$) = $50\text{m}$.
• Speed ($𝑣$) = $10\text{m/s}$.
• Centripetal acceleration (${𝑎}_{𝑐}$) = $\frac{{𝑣}^2}{𝑟}=\frac{{10}^2}{50}=2{\text{m/s}}^2$.

• Angular velocity ($𝜔$) = $4\text{rad/s}$.
• Radius ($𝑟$) = $2\text{m}$.
• Linear velocity ($𝑣$) = $𝑟\cdot 𝜔=2\cdot 4=8\text{m/s}$.

• Radius ($𝑟$) = $10\text{m}$.
• Centripetal acceleration (${𝑎}_{𝑐}$) = $16{\text{m/s}}^2$.
• Speed ($𝑣$) = $\sqrt{{𝑎}_{𝑐}\cdot 𝑟}=\sqrt{16\cdot 10}=\sqrt{160}=4\sqrt{10}\text{m/s}$.

• Mass ($𝑚$) = $0.5\text{kg}$.
• Radius ($𝑟$) = $2\text{m}$.
• Speed ($𝑣$) = $6\text{m/s}$.
• Centripetal force (${𝐹}_{𝑐}$) = $\frac{𝑚\cdot {𝑣}^2}{𝑟}=\frac{0.5\cdot 6^2}{2}=\frac{0.5\cdot 36}{2}=9\text{N}$.

• Time period ($𝑇$) = $4\text{s}$.
• Angular velocity ($𝜔$) = $\frac{2𝜋}{𝑇}=\frac{2\cdot \frac{22}{7}}{4}=\frac{44}{28}=1.57\text{rad/s}$.

Expected Questions

• Radius ($𝑟$) = $20\text{m}$.
• Speed ($𝑣$) = $5\text{m/s}$.
• Centripetal acceleration (${𝑎}_{𝑐}$) = $\frac{{𝑣}^2}{𝑟}=\frac{5^2}{20}=\frac{25}{20}=1.25{\text{m/s}}^2$.

• Angular velocity ($𝜔$) = $6\text{rad/s}$.
• Radius ($𝑟$) = $3\text{m}$.
• Linear velocity ($𝑣$) = $𝑟\cdot 𝜔=3\cdot 6=18\text{m/s}$.

• Radius ($𝑟$) = $40\text{m}$.
• Centripetal acceleration (${𝑎}_{𝑐}$) = $10{\text{m/s}}^2$.
• Speed ($𝑣$) = $\sqrt{{𝑎}_{𝑐}\cdot 𝑟}=\sqrt{10\cdot 40}=\sqrt{400}=20\text{m/s}$.

• Mass ($𝑚$) = $0.2\text{kg}$.
• Radius ($𝑟$) = $1.5\text{m}$.
• Speed ($𝑣$) = $3\text{m/s}$.
• Centripetal force (${𝐹}_{𝑐}$) = $\frac{𝑚\cdot {𝑣}^2}{𝑟}=\frac{0.2\cdot 3^2}{1.5}=\frac{0.2\cdot 9}{1.5}=\frac{1.8}{1.5}=1.2\text{N}$.

• Time period ($𝑇$) = $6\text{s}$.
• Angular velocity ($𝜔$) = $\frac{2𝜋}{𝑇}=\frac{2\cdot \frac{22}{7}}{6}=\frac{44}{42}=1.05\text{rad/s}$.