Question

A wheel rotates with a constant acceleration of $2.0 \mathrm{rad} / \mathrm{sec}^{2}$ If the wheel start from rest. The number of revolution it makes in the first ten seconds will be approximately. \(\\{\mathrm{A}\\} 8\) \\{B \\} 16 \(\\{\mathrm{C}\\} 24\) \(\\{\mathrm{D}\\} 32\)

Short answer

The number of revolutions made by the wheel in the first 10 seconds is approximately \(\\{B\\} 16\).

Step-by-step solution

Write down relevant equations of motion for constant angular acceleration

There are two main equations that we can use for this problem: 1. Angular displacement equation (final angle, initial angle, initial angular velocity, time, and angular acceleration): \(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\) 2. Angular velocity equation (final angular velocity, initial angular velocity, angular acceleration, and time): \(\omega = \omega_0 + \alpha t\) In this problem, we know that the initial angular velocity \(\omega_0 = 0\), the angular acceleration \(\alpha = 2.0 \, \text{rad/sec}^2\), and the time \(t = 10 \, \text{sec}\). We need to find the angular displacement in radians, \(\theta\), to calculate the number of revolutions.

Calculate the angular displacement using the first equation of motion

Using the equation \(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\), we can substitute our known values and calculate the angular displacement: \(\theta = 0 + 0 \cdot 10 + \frac{1}{2}(2.0) \cdot 10^2\) Calculating this value, we get: \(\theta = 100 \, \text{rad}\)

Convert angular displacement in radians to revolutions

Since there are 2π radians in one revolution, we can divide the angular displacement by 2π to get the number of revolutions: Number of revolutions = \(\frac{100 \, \text{rad}}{2π}\) Approximating π as 3.14, we have: Number of revolutions = \(\frac{100}{6.28} \approx 15.9\)

Choose the closest answer from the options

From our calculations, the number of revolutions is approximately 15.9. Looking at the answer choices, option B (16) is the closest to our calculated value. Therefore, the answer to this problem is: \(\\{B\\} 16\)