A solid cylinder with a radius of 5.08cm starts from rest at the top of a 12 meter long ramp inclined 20.3 degrees above the horizontal. When it reaches the bottom of the ramp 3.25 seconds later the cylinder has a final linear velocity of 7.38 m/sec. What was the average angular velocity of the cylinder?
To find the average angular velocity of the cylinder, we can use the equation:
Average angular velocity = change in angle / change in time
First, let's find the change in angle of the cylinder.
The cylinder rolls down the ramp without slipping, so the distance it travels along the ramp is equal to the arc length of the circle with radius 5.08cm.
Distance along the ramp = arc length of the circle
Distance along the ramp = radius * angle
Let's calculate the distance along the ramp:
Distance along the ramp = 12m * sin(20.3 degrees) (sin for 20.3 is 0.3502)
Distance along the ramp = 12m * 0.3502
Distance along the ramp = 4.2024m
Now, we know that the linear velocity of the cylinder at the bottom of the ramp is 7.38 m/sec. We can use this information to find the change in time.
Change in time = Distance along the ramp / Linear velocity at the bottom of the ramp
Change in time = 4.2024m / 7.38 m/sec
Change in time = 0.5696 sec
Now, we can calculate the change in angle using the formula:
Change in angle = Average angular velocity * Change in time
Let's rearrange the formula to solve for average angular velocity:
Average angular velocity = Change in angle / Change in time
Average angular velocity = 2π / (12m / 0.5696 sec)
Average angular velocity = 0.939 rad/sec
Therefore, the average angular velocity of the cylinder is 0.939 rad/sec.