Question

Grace is playing with her dolls and decides to give them a ride on a merry-go- round. She places one of them on an old record player turntable and sets the angular speed at 33.3 rpm. (a) What is their angular speed in rad/s? (b) If the doll is \(13 \mathrm{cm}\) from the center of the spinning turntable platform, how fast (in \(\mathrm{m} / \mathrm{s}\) ) is the doll moving?

Short answer

Answer: The doll's linear speed on the merry-go-round is approximately 0.453 m/s.

Step-by-step solution

(a) Convert Angular Speed from rpm to rad/s

First, we need to convert the given angular speed of 33.3 rpm (revolutions per minute) to radians per second (rad/s). We know that there are \(2\pi\) radians per revolution and 60 seconds per minute. So, we can use the following conversion formula: $$ \mathrm{angular\,speed\,in\,rad/s} = \mathrm{angular\,speed\,in\,rpm} \times \frac{2\pi\,\mathrm{rad}}{\mathrm{rev}} \times \frac{1\,\mathrm{min}}{60\,\mathrm{s}} $$ Now, substitute the given angular speed and calculate the value: $$ \mathrm{angular\,speed\,in\,rad/s} = 33.3\,\mathrm{rpm} \times \frac{2\pi\,\mathrm{rad}}{\mathrm{rev}} \times \frac{1\,\mathrm{min}}{60\,\mathrm{s}} $$

Calculation

$$ \mathrm{angular\,speed\,in\,rad/s} = 33.3 \times \frac{2\pi}{60} \approx 3.49\,\mathrm{rad/s} $$

(b) Calculate Linear Speed

To find the linear speed of the doll, we'll use the formula for the relationship between linear and angular speed: $$ \mathrm{linear\,speed} = \mathrm{angular\,speed} \times \mathrm{radius} $$ We're given the radius in centimeters, and we need the result in meters per second. Therefore, we must first convert the radius to meters: $$ 13\,\mathrm{cm} = 0.13\,\mathrm{m} $$ Now we can find the linear speed by plugging in the values: $$ \mathrm{linear\,speed} = 3.49\,\mathrm{rad/s} \times 0.13\,\mathrm{m} $$

Calculation

$$ \mathrm{linear\,speed} \approx 0.453\,\mathrm{m/s} $$ So, the doll's linear speed on the merry-go-round is approximately 0.453 m/s.