Angular Velocity and Angular Acceleration - UNSOLVED PRACTICE SET

Q7. Define angular velocity and angular acceleration. Write their SI units and dimensions.

Q8. A particle moves in a circle of radius 0.5 m with angular velocity 4 rad/s. Calculate its linear velocity and centripetal acceleration.

Q9. Distinguish between uniform circular motion and uniformly accelerated rotational motion. Give one example of each.

Q10. A wheel rotates with constant angular acceleration. Write the rotational kinematic equations analogous to linear motion equations.

Q11. The second hand of a clock is 10 cm long. Calculate the linear speed of its tip.

Q12. A disc rotates about its centre. Compare the linear velocities of a point on the rim and a point halfway between the centre and the rim.

Q13. Derive the relationship between linear and angular quantities for a particle in circular motion. Show that:

(i) v = rω

(ii) a_t = rα (tangential acceleration)

(iii) a_c = rω² = v²/r (centripetal acceleration)

(iv) a = √(a_t² + a_c²) (total acceleration)

Draw a diagram showing all these components and explain their directions.

Q14. A rigid body rotates about a fixed axis with angular acceleration α. Starting from rest at t = 0:

(i) Derive expressions for angular displacement θ, angular velocity ω, and the angle rotated in the nth second.

(ii) Compare these with the corresponding equations for linear motion.

(iii) A wheel starts from rest and accelerates uniformly at 2 rad/s². Calculate the angle rotated in the first 5 seconds and the angular velocity at t = 5 s.

(iv) How many revolutions does it complete in these 5 seconds?

Q15. A student observes a ceiling fan in the classroom. The fan has three blades, each 0.4 m long. It starts from rest and reaches a steady speed of 300 rpm in 10 seconds.

(i) Calculate the angular acceleration of the fan.

(ii) Calculate the total angle rotated during the acceleration phase.

(iii) Calculate the linear speed of the tip of a blade when the fan reaches steady speed.

(iv) Calculate the centripetal acceleration of the tip at steady speed.

(v) When the fan is switched off, it takes 15 seconds to come to rest. Calculate the angular deceleration.

Q16. A wheel of radius 0.3 m starts from rest and accelerates uniformly. After 4 seconds, its angular velocity is 20 rad/s.

(i) Calculate the angular acceleration.

(ii) Calculate the total angle rotated in 4 seconds.

(iii) Calculate the linear distance traveled by a point on the rim.

(iv) Calculate the tangential acceleration of a point on the rim.

(v) Calculate the total acceleration of a point on the rim at t = 4 s.

Q17. A disc of radius 0.2 m rotates about its centre with angular position given by θ = 2t³ – 3t² + 4t, where θ is in radians and t in seconds.

(i) Calculate the angular velocity at t = 2 s.

(ii) Calculate the angular acceleration at t = 2 s.

(iii) Calculate the linear velocity of a point on the rim at t = 2 s.

(iv) Calculate the tangential acceleration of a point on the rim at t = 2 s.

(v) At what time is the angular acceleration zero?

Q18. In a school sports day, a student throws a discus. The discus leaves her hand with a linear velocity of 20 m/s. Her arm length (from shoulder to hand) is 0.7 m, and she rotates through an angle of 270° during the throw.

(i) Calculate the average angular velocity during the throw if the throwing motion takes 0.3 s.

(ii) Calculate the angular acceleration, assuming uniform acceleration.

(iii) Calculate the centripetal acceleration of the discus just before release.

(iv) If the student increases her arm speed so that the discus is released with 25% more linear velocity, by what factor does the required angular velocity change?

(v) A second student with shorter arms (0.5 m) wants to achieve the same release velocity. What angular velocity must she achieve?