A 5.0 kg \rm kg, 52-cm \rm cm-diameter cylinder rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released.
a)What is the magnitude of the cylinder's initial angular acceleration
My work:
I found the moment of inertia from the parallel axis theorem:
I=I_cm+Md^2
I=(1/2)MR^2+Md^2
(plugging in the #'s you get...
I=0.507kgm^2
torque=Ix angular acceleration
I_grav=-Mgx_cm
don't know what my center of mass is?
How do I find the torque so I can isolate for angular acceleration?
Torque = Ix = rF
T = -mgx_cm
The x_cm is the distance it was moved. Since it is on the edge this would just be -r. Therefore,
Ix = mgr
x= (mgr/I)
x= [(5kg)(9.8 m/s^2)(0.26 m)] / (0.507 kgm^2)
x= 25.128 rad/s^2
To find the torque acting on the cylinder, we need to consider the gravitational force acting on the center of mass of the cylinder.
The torque due to the gravitational force can be calculated using the formula:
Torque = force × perpendicular distance from the axis of rotation
In this case, the force is the gravitational force acting on the center of mass of the cylinder, which is equal to the weight of the cylinder. The weight can be calculated using the formula:
Weight = mass × acceleration due to gravity
In this case, the acceleration due to gravity is approximately 9.8 m/s^2.
The perpendicular distance from the axis of rotation to the center of mass of the cylinder is the same as the radius of the cylinder, which is given as 52 cm.
So, the torque due to gravity is:
Torque_gravity = Weight × radius
Plugging in the values:
Torque_gravity = (mass × acceleration due to gravity) × radius
Torque_gravity = (5.0 kg) × (9.8 m/s^2) × (0.52 m)
Now that we have the torque, we can use the equation:
Torque = moment of inertia × angular acceleration
To isolate for the angular acceleration, we rearrange the equation:
Angular acceleration = Torque / moment of inertia
The moment of inertia is already given as 0.507 kgm^2. Plugging in the values for torque and moment of inertia:
Angular acceleration = Torque_gravity / moment of inertia
Now, simply substitute the calculated values into the equation and solve for the angular acceleration.