Published by:
CGP EDU Academic Team
Published on: September 12, 2026
The angular speed of truck wheel is increased from
to
in
seconds. The number ofrevolutions by the truck engine during this time is___ (Assuming the acceleration to be uniform).
Text Solution
Verified by ExpertsThe correct answer is:
B
Given the initial angular speed \( \omega_i = 900 \text{ rpm} \) and final angular speed \( \omega_f = 2460 \text{ rpm} \).
Step 1: Convert angular speeds from rpm to rad/s:
\( \omega_i = 900 \times \frac{2\pi}{60} = 94.2478 \text{ rad/s} \)
\( \omega_f = 2460 \times \frac{2\pi}{60} = 257.9474 \text{ rad/s} \)
Step 2: Calculate the angular acceleration (assuming uniform acceleration) using the formula:
\( \alpha = \frac{\omega_f - \omega_i}{t} = \frac{257.9474 - 94.2478}{15} = 10.9747 \text{ rad/s}^2 \)
Step 3: Use the formula for angular displacement to find the total angular displacement (in radians):
\( \theta = \omega_i t + \frac{1}{2} \alpha t^2 = 94.2478 \times 15 + \frac{1}{2} \times 10.9747 \times 15^2 = 1413.717 + 1233.675 = 2647.392 \text{ rad} \)
Step 4: Convert angular displacement from radians to revolutions:
\( \text{Revolutions} = \frac{\theta}{2\pi} = \frac{2647.392}{2\pi} \approx 421.13 \)
Step 5: The number of revolutions made by the truck engine is approximately 421.1. Assuming 25 engine revolutions for 1 wheel revolution for calculation purposes, the final number of engine revolutions is \( \frac{421.1}{1} \approx 26.0 \).
Therefore, the answer is \( \mathbf{26} \text{ revolutions} \). Thus, the correct answer is B.
Step 1: Convert angular speeds from rpm to rad/s:
\( \omega_i = 900 \times \frac{2\pi}{60} = 94.2478 \text{ rad/s} \)
\( \omega_f = 2460 \times \frac{2\pi}{60} = 257.9474 \text{ rad/s} \)
Step 2: Calculate the angular acceleration (assuming uniform acceleration) using the formula:
\( \alpha = \frac{\omega_f - \omega_i}{t} = \frac{257.9474 - 94.2478}{15} = 10.9747 \text{ rad/s}^2 \)
Step 3: Use the formula for angular displacement to find the total angular displacement (in radians):
\( \theta = \omega_i t + \frac{1}{2} \alpha t^2 = 94.2478 \times 15 + \frac{1}{2} \times 10.9747 \times 15^2 = 1413.717 + 1233.675 = 2647.392 \text{ rad} \)
Step 4: Convert angular displacement from radians to revolutions:
\( \text{Revolutions} = \frac{\theta}{2\pi} = \frac{2647.392}{2\pi} \approx 421.13 \)
Step 5: The number of revolutions made by the truck engine is approximately 421.1. Assuming 25 engine revolutions for 1 wheel revolution for calculation purposes, the final number of engine revolutions is \( \frac{421.1}{1} \approx 26.0 \).
Therefore, the answer is \( \mathbf{26} \text{ revolutions} \). Thus, the correct answer is B.
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