Questions: A grinding wheel 0.31 m in diameter rotates at 2400 rpm Part A Calculate its angular velocity in rad/s. Express your answer using three significant figures. ω= rad / s Part B What is the linear speed of a point on the edge of the grinding wheel? Express your answer to two significant figures and include the appropriate units. v= Value Units

Solution Steps

The angular velocity \(\omega\) in radians per second can be calculated from the rotational speed in revolutions per minute (rpm) using the formula:

\[ \omega = \frac{2\pi \times \text{rpm}}{60} \]

Given that the grinding wheel rotates at 2400 rpm, we substitute this value into the formula:

\[ \omega = \frac{2\pi \times 2400}{60} \]

Calculate the angular velocity:

\[ \omega = \frac{2\pi \times 2400}{60} = 80\pi \approx 251.3274 \, \text{rad/s} \]

Rounding to three significant figures, we have:

\[ \omega \approx 251 \, \text{rad/s} \]

The linear speed \(v\) of a point on the edge of the wheel is given by the formula:

\[ v = r \times \omega \]

where \(r\) is the radius of the wheel. The diameter of the wheel is 0.31 m, so the radius \(r\) is:

\[ r = \frac{0.31}{2} = 0.155 \, \text{m} \]

Substitute the values of \(r\) and \(\omega\) into the formula:

\[ v = 0.155 \times 251.3274 \approx 38.4567 \, \text{m/s} \]

Rounding to two significant figures, we have:

\[ v \approx 38 \, \text{m/s} \]

Final Answer