Uniform Circular Motion: Comprehensive NEET Physics Notes

1. Uniform Circular Motion (UCM)

Uniform Circular Motion (UCM) occurs when an object moves in a circular path with constant speed but continuously changing velocity due to its changing direction. Even though the speed is constant, the object's direction is always changing, which gives rise to acceleration.

1.1 Angular Displacement

Angular displacement refers to the angle covered by an object moving along a circular path, measured in radians.

• Formula: $\theta =\frac{s}{r}$
• • where:
  • • $\theta$ is the angular displacement (radians),
    • $s$ is the arc length,
    • $r$ is the radius of the circular path.

Did You Know?

One complete revolution corresponds to an angular displacement of $2\pi$ radians, which is equivalent to 360 degrees.

1.2 Angular Velocity and Frequency

Angular velocity ($\omega$) represents how fast the object is rotating in the circular path. It is the rate of change of angular displacement with time.

• Formula: $\omega =\frac{d\theta }{dt}$
  For uniform motion: $\omega =\frac{2\pi }{T}=2\pi f$
• • where:
  • • $T$ is the time period of one revolution,
    • $f$ is the frequency of revolutions (in Hz).

The relation between angular velocity and linear velocity ($v$) is: $v=r\omega$

NEET Tip:

When tackling NEET problems, distinguish between angular velocity and linear velocity. Use the appropriate formula based on the problem context.

Real-life Application:

A rotating fan demonstrates UCM. Each blade moves in a circular path with constant speed, but its direction is continuously changing. The faster the fan rotates, the greater the angular velocity.

1.3 Centripetal Acceleration

In UCM, even though the object moves with constant speed, it experiences a continuous inward acceleration known as centripetal acceleration ($a_c$), which is directed toward the center of the circular path.

• Formula: $a_c=\frac{v^2}{r}=r{\omega }^2$
• • where:
  • • $v$ is the linear velocity,
    • $r$ is the radius of the path,
    • $\omega$ is the angular velocity.

Mnemonic:

"ACCelerate INside" - Centripetal ACCeleration is always directed INside the circular path.

Common Misconception:

Centrifugal force is often misunderstood as a real force. In reality, it's a pseudo-force felt by an object moving in a rotating frame. The actual force responsible for keeping an object in a circular path is centripetal force, directed inward.

1.4 Centripetal Force

Centripetal force is the inward force that keeps an object moving along a circular path. It prevents the object from moving in a straight line by constantly pulling it toward the center of the circle.

• Formula: $F_c=\frac{m{v}^2}{r}=mr{\omega }^2$
• • where:
  • • $F_c$ is the centripetal force,
    • $m$ is the mass of the object,
    • $v$ is the linear velocity,
    • $r$ is the radius of the circular path.

Real-life Application:

When a car makes a sharp turn, the friction between the tires and the road provides the centripetal force, preventing the car from skidding outward. Without sufficient friction (like on a slippery road), the car may lose control.

Quick Recap:

• Uniform Circular Motion: Motion along a circular path at constant speed but varying velocity.
• Angular Displacement: $\theta =\frac{s}{r}$, measured in radians.
• Angular Velocity: $\omega =\frac{d\theta }{dt}$ or $\omega =\frac{2\pi }{T}$.
• Centripetal Acceleration: $a_c=\frac{v^2}{r}=r{\omega }^2$.
• Centripetal Force: $F_c=\frac{m{v}^2}{r}=mr{\omega }^2$.

Practice Questions:

1. A car moves at a constant speed of 15 m/s along a circular track of radius 40 m. Calculate its centripetal acceleration.
   Solution: $a_c=\frac{v^2}{r}=\frac{(15{)}^2}{40}=5.625{\text{m/s}}^2$
2. An object of mass 3 kg is moving in a circular path of radius 0.4 m with an angular velocity of 8 rad/s. Calculate the centripetal force.
   Solution: $F_c=mr{\omega }^2=3\times 0.4\times (8{)}^2=76.8\text{N}$
3. A satellite completes one orbit around the Earth in 2 hours. Calculate its angular velocity in rad/s.
   Solution: Time period $T=2\times 60\times 60=7200\text{seconds}$ $\omega =\frac{2\pi }{T}=\frac{2\pi }{7200}\approx 0.00087\text{rad/s}$
4. An object completes 7 revolutions per second. Calculate its angular velocity.
   Solution: $\omega =2\pi f=2\pi \times 7=43.98\text{rad/s}$
5. A ball of mass 1 kg tied to a string moves in a circular path with a radius of 0.5 m and a speed of 4 m/s. Calculate the centripetal force.
   Solution: $F_c=\frac{m{v}^2}{r}=\frac{1\times (4{)}^2}{0.5}=32\text{N}$

NEET Exam Strategy:

1. Memorize Key Formulas: Ensure you know all the important formulas for angular velocity, centripetal force, and acceleration.
2. Focus on Problem-Solving: Practice questions involving centripetal force and acceleration frequently appear in NEET exams.
3. Watch for Units: Be mindful of units when solving problems. Always convert time to seconds and distance to meters when needed.

Improvements Based on Previous Feedback:

• Visual Aids: Diagrams of circular motion, showing forces and velocity vectors, should be incorporated to improve conceptual understanding. For instance, a diagram showing centripetal force acting inward on a car turning a corner would illustrate the concept effectively.
  Example of a helpful diagram:
• • A diagram showing a car making a turn with forces like friction (providing centripetal force) clearly labeled.
• Additional NEET-style Practice Questions: Incorporating more varied NEET-style questions, including those that require step-by-step calculation, will help students familiarize themselves with potential exam questions.
• Glossary and Quick Reference: A glossary section for key terms such as angular velocity, centripetal force, and angular displacement would aid in revision. Including a quick reference guide for formul