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 cm from the center of the spinning
turntable platform, how fast (in m/s) is the doll moving?
omega = (33.3 revs / 60 seconds) * ( 2 * 3.14 radians /rev) = 3.49 radians/second
R = 0.13 meter
v = omega * R = 3.49 * 0.13 = 0.453 m/s
Well, let me put on my jester hat and calculate that for you!
(a) To convert from rpm to rad/s, we need to multiply the angular speed by 2π (since there are 2π radians in a full revolution). So, 33.3 rpm multiplied by 2π equals approximately 208.956 rad/s.
(b) Now, to find how fast the doll is moving, we'll use the formula v = ωr, where v is the linear speed, ω is the angular speed, and r is the radius.
Given that the doll is 13 cm (or 0.13 m) from the center, we can plug in the values:
v = 208.956 rad/s multiplied by 0.13 m
Calculating that gives us approximately 27.1736 m/s.
So, the doll is movin' and groovin' at around 27.1736 m/s! Just don't let it get too dizzy!
To find the answers, we need to convert the given angular speed from rpm to rad/s and then calculate the linear speed of the doll.
(a) To convert from rpm (revolutions per minute) to rad/s (radians per second), we can use the following equation:
Angular speed in rad/s = (Angular speed in rpm) * (2π rad/1 revolution) * (1 min/60 s)
Given that the angular speed is 33.3 rpm, we can calculate it in rad/s as follows:
Angular speed in rad/s = 33.3 rpm * (2π rad/1 revolution) * (1 min/60 s)
Angular speed in rad/s = (33.3 * 2π) / 60 rad/s
Calculating this expression, we get:
Angular speed in rad/s ≈ 3.49 rad/s
Therefore, the angular speed of the dolls on the turntable is approximately 3.49 rad/s.
(b) To find the linear speed of the doll, we can use the formula:
Linear speed = (Radius) * (Angular speed)
Given that the doll is 13 cm (or 0.13 m) from the center of the turntable platform, and we have already calculated the angular speed as 3.49 rad/s, we can calculate the linear speed as follows:
Linear speed = 0.13 m * 3.49 rad/s
Linear speed ≈ 0.45 m/s
Therefore, the doll is moving at a speed of approximately 0.45 m/s.
To answer both parts of the question, we need to convert the angular speed from revolutions per minute (rpm) to radians per second (rad/s). We also need to calculate the linear speed of the doll on the turntable.
(a) To convert the angular speed from rpm to rad/s, we need to use the conversion factor:
1 revolution = 2π radians
So, to get the angular speed in rad/s, we can multiply the angular speed in rpm by the conversion factor:
Angular speed in rad/s = 33.3 rpm * (2π radians/1 revolution) = 33.3 * 2π rad/min
Now, we need to convert the angular speed from rad/min to rad/s. Since there are 60 seconds in a minute, we divide the angular speed by 60 to get the result in rad/s:
Angular speed in rad/s = (33.3 * 2π rad/min) / 60 = 3.49 rad/s (approximately)
Therefore, the angular speed of the doll on the turntable is approximately 3.49 rad/s.
(b) To calculate the linear speed of the doll, we use the formula:
Linear speed = angular speed * radius
Given that the doll is 13 cm (0.13 m) from the center of the turntable, we can calculate the linear speed:
Linear speed = 3.49 rad/s * 0.13 m = 0.4537 m/s (approximately)
Therefore, the doll is moving at a speed of approximately 0.4537 m/s on the spinning turntable.