A particle moves in a circular path of 0.3 m radius. Calculate the magnitude a of the acceleration of the particle in m/s2 (a) if its speed is constant at 0.6 m/s and (b) if its speed is 0.6 m/s but is increasing at the rate of 0.9 m/s .

Question

A particle moves in a circular path of 0.3 m radius. Calculate the magnitude a of the acceleration of the particle in m/s2  (a) if its speed is constant at 0.6 m/s and (b) if its speed is 0.6 m/s but is increasing at the rate of 0.9 m/s .

BlackTom AI · Accepted Answer

The problem asks for the magnitude of the acceleration of a particle moving in a circle of radius 0.3 m under two scenarios: (a) constant speed 0.6 m/s, (b) speed 0.6 m/s with a tangential acceleration of 0.9 m/s^2. For circular motion with constant speed, the acceleration is purely centripetal, given by a_c = v^2 / r. For non-constant speed, the total acceleration combines centripetal and tangential components: a = sqrt(a_c^2 + a_t^2), where a_t is the tangential acceleration dv/dt.

Option 1: (a) 1.2 (b) 1.9
- In scenario (a), compute a_c = v^2 / r = (0.6)^2 / 0.3 = 0.36 / 0.3 = 1.2 m/s^2, which matches the first value here. For scenario (b), this option would imply a resulting magnitude of 1.9 m/s^2. If we combine a_c = 1.2 with a_t = 0.9 using a = sqrt(a_c^2 + a_t^2), we get sqrt(1.44 + 0.81) = sqrt(2.25) = 1.5 m/s^2, not 1.9. So this option fails the second part.

Option 2: (a) 1.8 (b) 1.9
- The centripetal component a_c with v = 0.6 and r = 0.3 remains 1.2 m/s^2, not 1.8, so the first part is inconsistent with the given data. Even without checking the second part, this already indicates a mismatch with (a). For (b), the combination would still yield 1.5 m/s^2 as shown earlier, not 1.9. Overall, this option is incorrect due to both parts not aligning with the calculated values.

Option 3: (a) 1.2 (b) 1.5
- For (a), a_c = v^2 / r = (0.6)^2 / 0.3 = 1.2 m/s^2, which is consistent with this option’s first value. For (b), tangential acceleration a_t = 0.9 m/s^2 and centripetal a_c = 1.2 m/s^2. The total acceleration magnitude is a = sqrt(a_c^2 + a_t^2) = sqrt(1.44 + 0.81) = sqrt(2.25) = 1.5 m/s^2, matching the second value shown. This option aligns with both parts of the problem.

Option 4: (a) 1.6 (b) 1.5
- The first part would require a_c = 1.6 m/s^2, which would demand v^2 / r = 1.6. With r = 0.3, that would require v^2 = 0.48, so v ≈ 0.693 m/s, which disagrees with the given speed of 0.6 m/s. Hence (a) is inconsistent. The second part, while it coincidentally yields 1.5 m/s^2, relies on an incorrect centripetal value from (a), so this option cannot be correct overall.

In summary, evaluate each option by checking the centripetal calculation for part (a) and, if speed changes in part (b), combining centripetal and tangential components as a = sqrt(a_c^2 + a_t^2) to verify the (b) value. The only choice that satisfies both parts consistently with the given data is the one where (a) matches 1.2 m/s^2 and (b) matches 1.5 m/s^2.

MIET2340 Quiz (10%)

A particle moves in a circular path of 0.3 m radius. Calculate the magnitude a of the acceleration of the particle in m/s2 (a) if its speed is constant at 0.6 m/s and (b) if its speed is 0.6 m/s but is increasing at the rate of 0.9 m/s .

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