More practice questions

4. Q4. The second hand of a clock is 2.0 cm long. Find the speed of its tip and its centripetal acceleration.

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   T = 60 s, so ω = 2π/60 ≈ 0.105 rad/s. v = ωR ≈ 0.105 × 2.0 ≈ 0.21 cm/s; a = ω^²R ≈ 0.011 × 2.0 ≈ 0.022 cm s^⁻².

5. Q5. Two particles move on circles of radii 1 m and 4 m with the same period. Compare their speeds and centripetal accelerations.

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   Same T means the same ω. v = ωR, so the speeds are in the ratio 1 : 4; a = ω^²R, so the accelerations are also 1 : 4.

6. Q6. A particle moves on a circle of radius 2 m at a constant 4 m/s. Find the magnitude of its average acceleration over a quarter of a revolution.

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   The velocity turns through 90° with no change in size, so |Δv| = 4√2 ≈ 5.66 m/s. The time is Δt = T/4, with T = 2πR/v = 2π × 2/4 = π s, so Δt = π/4 ≈ 0.785 s. Average acceleration ≈ 5.66/0.785 ≈ 7.2 m s^⁻², a little less than the instantaneous v^²/R = 8 m s^⁻².