MIT
A uniform disc of mass m and radius r is acted upon by three forces. find the magnitude of angular acceleration and the sense of rotation. Three different cases. force F, 3F, 2F. need help, not doing so good.
This makes no sense.
remember that the sum of the torques is equal to I*alpha
so sigma(t) = r1xF1 + r2xF2 +... +rnxFn = I*alpha
where I is the moment of inertia through the axis of rotation. look for I in a table, easily found on google. notice the similarity to the application of Newtons second law.
set up your problem like that and then just solve for alpha.
remember to set signs for positive and negative torques, i like sticking to a positive CCW, so that when all the analysis is done, the direction of rotation resolves with the sign automatically.
since the moment of inertia depends on the axis of rotation, part b requires the parallel axis theorem to get the moment of inertia through the new axis of rotation, although some tables include the moment of inertia you're looking for already calculated for you.
watch out for the fact that if a force's line of action intersects with the axis of rotation, the force will not cause a torque because r=0 so rxF=0. Remember this paragraph while setting up part B
once you've found your new moment of inertia. the same idea that worked for part A will help you solve for alpha in part B.
part C is a bit more laborious but if you solved A and B correctly, it's just more of the same. as opposed to part B, all three forces provide a torque.
all three forces are applied tangentially to their Rs in all three cases so theres no need for the cross product in any part of this problem.
so, algorithmically:
1.- find out the pertinent moment of inertia
2.- set up your problem as following:
sigma(t)= I*alpha = (r1 x F1) + (r2 x F2) +... + (rn x Fn)
3.- get your alpha algebraically.
if the algebra messes you up, google wolfram alpha
Good luck
a) alpha= 2F/mr ccw
b) alpha= 2F/mr cw
c) alpha= 8F/3mr ccw
To find the magnitude of the angular acceleration and the sense of rotation for the given cases, we can use Newton's second law for rotation, which states that the net torque acting on an object is equal to the moment of inertia times the angular acceleration.
The moment of inertia (I) for a uniform disc rotating about its center can be given by the equation:
I = (1/2) * m * r^2
where m is the mass of the disc and r is its radius.
Let's analyze each case separately:
Case 1: Force F
In this case, the force acting on the disc is F. Since the force is applied at the center of the disc, it does not create any torque and, therefore, the disc will not experience any angular acceleration or rotation.
Case 2: Force 3F
Here, the force acting on the disc is 3F. Since the force is still applied at the center of the disc, there is no torque again, resulting in no angular acceleration or rotation.
Case 3: Force 2F
In this case, the force acting on the disc is 2F. This time, the force is not applied at the center, but rather at some distance from the center. To find the torque created by this force, we can use the formula:
τ = r * F
where τ is the torque, r is the distance from the center of the disc to the point where the force is applied, and F is the magnitude of the force.
Since the two forces of 2F and F are applied at mirror positions on opposite sides of the disc and have the same distance from the center, the net torque can be calculated as:
τ(net) = (2F) * r - (F) * r = rF
Now, applying Newton's second law for rotation, we have:
τ(net) = I * α
where α is the angular acceleration.
Substituting the values of I and τ(net) in the equation, we get:
rF = (1/2) * m * r^2 * α
Simplifying the equation:
α = (2F) / (m * r)
To determine the sense of rotation, we need additional information about the orientation of the force relative to the initial orientation of the disc. The sense of rotation can be determined by using the right-hand rule, where you align your thumb with the direction of the force and observe the curling direction of your fingers.
Remember to plug in the appropriate values for the mass (m), radius (r), and force magnitude (F) to calculate the angular acceleration (α) for the given case.