A circular pulley with a radius of 10 cm is turning at 12 revolutions per minute. How
fast is a point on the edge of the pulley rising when it is 5 cm higher than the center of
the pulley?
To find the speed at which a point on the edge of the pulley is rising, we can use the concept of angular velocity.
Given:
Radius of the pulley, r = 10 cm
Revolutions per minute, ω = 12 rpm
Height of the point above the center, h = 5 cm
Now, let's determine the angular velocity (ω) in radians per second:
1 revolution = 2π radians
12 revolutions per minute = 12 * 2π radians per minute
ω = (12 * 2π) / 60 = 4π / 5 radians per second
Next, we can use the relationship between linear velocity (v) and angular velocity (ω) for a point on the edge of the pulley. The linear velocity of a point on the edge of a pulley can be found using the formula:
v = ω * r
Substituting the given values:
v = (4π / 5) * 10
v = 8π cm/s
Therefore, the point on the edge of the pulley is rising at a speed of 8π cm/s when it is 5 cm higher than the center of the pulley.
To find the speed at which a point on the edge of the pulley is rising, we can use the concept of angular velocity and relate it to linear velocity.
First, let's calculate the angular velocity (ω) of the pulley. Given that the pulley is turning at 12 revolutions per minute, we can convert this to radians per minute using the conversion factor 1 revolution = 2π radians. Therefore:
Angular velocity (ω) = 12 revolutions/minute * 2π radians/revolution
Next, we need to find the linear velocity (v) of the point on the edge of the pulley. The linear velocity of a point on the edge of a rotating object can be calculated by the formula:
Linear velocity (v) = Angular velocity (ω) * Radius (r)
Since we are given that the radius (r) of the pulley is 10 cm, we substitute the values and solve for the linear velocity:
Linear velocity (v) = ω * r
Now, we have the linear velocity of the point on the edge of the pulley. However, we need to find the rate at which it is rising when it is 5 cm higher (above) the center of the pulley.
To do this, we need to consider the relationship between linear velocity and the rate of rise. Since the linear velocity is perpendicular to the direction of rise, we can apply Pythagoras' theorem to find the total velocity (V) of the point:
Total velocity (V) = square root of (Linear velocity (v)^2 + Rate of rise (h)^2)
We are given that the rate of rise (h) is 5 cm. Substituting the values, we can solve for the total velocity:
Total velocity (V) = square root of (v^2 + h^2)
Finally, we have the total velocity of the point on the edge of the pulley.