A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch. The small wheel has a radius of 1.7 cm and accelerates at the rate of 6.9 rad/s^2, and it is in contact with the pottery wheel (radius 27.0 cm) without slipping.
(1) Calculate the angular acceleration of the pottery wheel.
(2) Calculate the time it takes the pottery wheel to reach its required speed of 59 rpm.
Since the velocities of the rims are equal at the point of contact, the angular velocities are inversely proportional to the radii. The same applies to the angular acceleration rates.
use that fact to answer the questions yourself
To solve this problem, we can use the concept of rotational motion and the relationship between the angular acceleration, radius, and linear acceleration for objects in contact without slipping.
(1) To calculate the angular acceleration of the pottery wheel, we need to use the relationship between the linear acceleration and the angular acceleration.
The linear acceleration of the small wheel is given as 6.9 rad/s^2, which is the same as the angular acceleration since the radius of the small wheel is known (1.7 cm). The angular acceleration of the pottery wheel can be found using the ratio of the radii of the two wheels.
Angular acceleration of pottery wheel = (Radius of small wheel / Radius of pottery wheel) * Linear acceleration of small wheel
Angular acceleration = (1.7 cm / 27.0 cm) * 6.9 rad/s^2
Calculating the value, we get:
Angular acceleration = (0.06296) * 6.9 rad/s^2
Angular acceleration = 0.4333 rad/s^2
So, the angular acceleration of the pottery wheel is approximately 0.4333 rad/s^2.
(2) To calculate the time it takes the pottery wheel to reach its required speed of 59 rpm, we can use the formula:
Angular speed = Angular acceleration * time
The angular speed is given in revolutions per minute (rpm). To convert this to radians per second (rad/s), we use the conversion factor:
1 revolution = 2π radians
So, 59 rpm = (59 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds)
59 rpm = 6.1831 rad/s
Now, we can rearrange the formula to calculate the time:
Time = Angular speed / Angular acceleration
Time = 6.1831 rad/s / 0.4333 rad/s^2
Calculating the value, we get:
Time = 14.28 seconds
Therefore, it takes approximately 14.28 seconds for the pottery wheel to reach its required speed of 59 rpm.