What is the angular momentum of a 2.4 kg uniform cylindrical grinding wheel of radius 12 cm when rotating at 1800 rpm?
How much torque is required to stop it in 4.0 s?
Convert 1800 rpm to radians per second and call it w. (It texts, it's usually a Greek lower case omega)
The angular momentum is I w, where I is the moment of inertia. Use units of kg m^2/s
For a solid cylinder of mass M and radius R, I = (1/2) M R^2.
The torque L required to stop in time T is I w/T. The units will be newton meters or kg m^2/s^2 (same thing)
To calculate the angular momentum of an object, we use the formula:
Angular momentum (L) = moment of inertia (I) x angular velocity (ω)
To find the moment of inertia (I) of the cylindrical grinding wheel, we can use the formula for the moment of inertia of a uniform cylinder:
I = 1/2 * m * r^2
Where:
m = mass of the grinding wheel
r = radius of the grinding wheel
Given that the mass of the grinding wheel is 2.4 kg and the radius is 12 cm (which is equal to 0.12 m), we can calculate the moment of inertia (I):
I = 1/2 * 2.4 kg * (0.12 m)^2
Next, we need to convert the angular velocity from revolutions per minute (rpm) to radians per second (rad/s). Since 1 revolution is equal to 2π radians, we can use the conversion factor:
1 rpm = (2π/60) rad/s
Given that the angular velocity is 1800 rpm, we can calculate it in radians per second:
ω = 1800 rpm * (2π/60) rad/s
Finally, we can calculate the angular momentum (L) using the formula mentioned earlier:
L = I * ω
For the torque required to stop the grinding wheel, we can use the formula:
Torque (τ) = change in angular momentum / time interval
Given that the time interval is 4.0 s, we can calculate the torque:
τ = (L - 0) / 4.0 s
where 0 represents the final angular momentum, as the grinding wheel needs to be stopped.
By following these steps, you can calculate both the angular momentum and the torque required to stop the grinding wheel in 4.0 s.