More practice questions

4. Q4. Blocks of 3 kg and 2 kg hang on either side of a string over a pulley that is a uniform disc of mass 2 kg, turning without friction; the string does not slip. Find the acceleration of the blocks. Take g = 9.8 m s^⁻².

   Show answerHide answer

   With tensions T₁ (3 kg side) and T₂ (2 kg side): 3g − T₁ = 3a, T₂ − 2g = 2a, and (T₁ − T₂)R = (MR^²/2)(a/R), so T₁ − T₂ = a. Adding all three: 3g − 2g = (3 + 2 + 1)a, so g = 6a and a = 9.8/6, about 1.63 m s^⁻². The tensions differ because the pulley needs a net torque to speed up.

5. Q5. In Example 3, the block starts from rest. Use energy to find its speed after it has fallen 1 m.

   Show answerHide answer

   The loss of potential energy mgh becomes ½mv^² + ½Iω^², with ω = v/R and I = MR^²/2, so mgh = ½(m + M/2)v^². Then v^² = 2 × 2 × 9.8 × 1/(2 + 2) = 9.8, and v is about 3.13 m s^⁻¹. (Check: v^² = 2ah = 2 × 4.9 × 1 = 9.8.)

6. Q6. A record-player turntable with I = 0.2 kg m^² is driven from rest by a constant motor torque of 0.5 N m, with bearing friction neglected. How long does it take to reach 100/3 rpm, and how much work does the motor do?

   Show answerHide answer

   100/3 rpm is 2π × (100/3)/60, about 3.49 rad s^⁻¹. α = 0.5/0.2 = 2.5 rad s^⁻², so t = 3.49/2.5, about 1.40 s. The work is ½ × 0.2 × 3.49^², about 1.22 J.