Question

A flywheel of radius \(27.01 \mathrm{~cm}\) rotates with a frequency of \(4949 \mathrm{rpm} .\) What is the centripetal acceleration at a point on the edge of the flywheel?

Short answer

Answer: The centripetal acceleration at a point on the edge of the flywheel is approximately 72859.21 m/s².

Step-by-step solution

Convert frequency to angular velocity

We have the frequency in rpm, so first, we will convert it to radians per second. To do this, we will use the conversion factor \(\frac{2\pi}{60}\), as there are \(2\pi\) radians in one rotation and 60 seconds in one minute. The formula for converting frequency to angular velocity is: \(\omega = f \times \frac{2\pi}{60}\) Given frequency, \(f = 4949 \,\text{rpm}\), let's calculate the angular velocity: \(\omega = 4949 \times \frac{2\pi}{60}\)

Calculate angular velocity

Now, we can plug in the value of \(f\) and calculate \(\omega\): \(\omega = 4949 \times \frac{2\pi}{60} = 518.34 \,\text{rad/s}\) We have now converted the given frequency to angular velocity in radians per second.

Find the centripetal acceleration

Now that we have the angular velocity, we can use it to find the centripetal acceleration using the formula \(a_c = R\omega^2\). Given the radius, \(R = 27.01 \,\text{cm}\), we can find the centripetal acceleration as follows: First, we need to convert \(R\) from centimeters to meters: \(R = 27.01 \,\text{cm} \times \frac{1 \,\text{m}}{100 \,\text{cm}} = 0.2701 \,\text{m}\) Now, we can calculate the centripetal acceleration: \(a_c = R\omega^2 = 0.2701 \times (518.34)^2\)

Calculate centripetal acceleration

Plugging in the values of \(R\) and \(\omega\), we can find \(a_c\): \(a_c = 0.2701 \times (518.34)^2 = 72859.21 \, \text{m/s}^2\) The centripetal acceleration at a point on the edge of the flywheel is approximately \(72859.21 \, \text{m/s}^2\).