How much tangential force must be exerted to stop a rotating solid disk (radius = 1.50 feet) weighing 480 lbs in 12.0 seconds? The disk is initially rotating at 600 RPM.
aarrgh, English units
torque = F * .75 ft = .75 F foot pounds
mass = 480/32.2 = 14.9 slugs
I = (1/2) m r^2 = .5 (14.9)(.5625)
= 4.19 slug ft^2
kinetic energy = (1/2) I w^2
find w in radians/sec
w = omega = 600 rev/60 s * 2 pi rad/rev = 62.8 rad/s
so
ke = (1/2)(4.19)(62.8)^2 = 8271 slug ft^2/s^2
work done by force = torque * total angle in radians to stop
= .75 F (theta)
theta is average speed in rad/s times time to stop
average speed = 62.8/2 = 31.4 rad/s
so
theta = 31.4 rad/s * 12 s
= 377 radians to stop
so
work to stop = .75 F (377)
= 283 F ft pounds
so
283 F = 8271
F = 29.2 pounds
I used radius = .75, should be 1.5
aarrgh, English units
torque = F * 1.5 ft = 1.5 F foot pounds
mass = 480/32.2 = 14.9 slugs
I = (1/2) m r^2 = .5 (14.9)(2.25)
= 16.8 slug ft^2
kinetic energy = (1/2) I w^2
find w in radians/sec
w = omega = 600 rev/60 s * 2 pi rad/rev = 62.8 rad/s
so
ke = (1/2)(16.8)(62.8)^2 = 33,054 slug ft^2/s^2
work done by force = torque * total angle in radians to stop
= 1.5 F (theta)
theta is average speed in rad/s times time to stop
average speed = 62.8/2 = 31.4 rad/s
so
theta = 31.4 rad/s * 12 s
= 377 radians to stop
so
work to stop = 1.5 F (377)
= 566 F ft pounds
so
566 F = 33,054
F = 58.5 pounds
To determine the tangential force required to stop a rotating disk, we can use the principle of rotational motion. The formula to calculate the tangential force is:
Tangential force = Moment of inertia x Angular acceleration
To find the moment of inertia of a solid disk, we use the formula:
Moment of inertia = (1/2) x mass x radius^2
First, we need to find the mass of the disk. Since the weight is given in pounds, we need to convert it to mass in pounds (lb) by dividing it by the acceleration due to gravity, which is approximately 32.2 ft/s^2.
Mass = Weight / Acceleration due to gravity = 480 lb / 32.2 ft/s^2
Next, we can calculate the moment of inertia:
Moment of inertia = (1/2) x mass x radius^2
Now, we need to find the angular acceleration. We can use the formula:
Angular acceleration = Change in angular velocity / Time
The initial angular velocity is given as 600 RPM (revolutions per minute). To convert it to radians per second (rad/s), we multiply by 2π/60.
Change in angular velocity = 0 - initial angular velocity
Finally, we can calculate the tangential force using the formula:
Tangential force = Moment of inertia x Angular acceleration
By plugging in the given values and solving the equations, we can find the answer.