2. [1pt] A uniform 3.9 kg cylinder can rotate about an axis through its center a

Question

2. [1pt]
A uniform 3.9 kg cylinder can rotate about an axis through its center at O. The forces applied are: F1 = 7.80 N, F2 = 4.00 N, F3 = 4.40 N, and F4 = 3.80 N. Also, R1 = 11.0 cm and R3 = 4.70 cm. Find the angular acceleration of the cylinder.

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3. [1pt]
A pulley with mass Mp and radius Rp is attached to the ceiling and rotates with no friction about its pivot. Two masses are hung from the ends of a massless string looped over the pulley; one mass, M2, is larger than the other mass, m1. The magnitudes of the various tensions are Tn; the magnitude of the gravitational acceleration is g. For each of the statements below, select the appropriate answer from T-True, F-False, G-Greater than, L-Less than, E-Equal to. E.g., if the first is G and the rest, L, enter GLLLLL.

i) The center of mass of Mp + m1 + M2 accelerates.

ii) m1g + M2g + Mpg is ____ T3.

iii) The magnitude of the acceleration of M2 is ____ that of m1.

iv) T3 is ____ T1+T2

v) T1 is ____ T2

vi) T2 is ____ M2g.

Explanation / Answer

There is an equation similar to "F=ma" that deals with angular motion. But instead of force (F), it uses torque (); instead of mass (m), it uses "moment of inertia" (I); and instead of acceleration (a), it uses "angular acceleration" ().

Torque = (moment of inertia) × (angular acceleration)

= I

Torque equals (force)×(lever arm). The "lever arm" is the perpendicular separation between the line of force and the axis.

In the diagram, there are four forces, therefore four torques. You need to add them all up to get the total torque ()

Torque_1 = F1 × R (positive because it pulls counter-clockwise). ...7.8*.11=0.858
Torque_2 = F2 × R (negative because it pulls clockwise) ........4*.11=0.44
Torque_3 = F3 × r (negative because it pulls clockwise). ..............4.4*0.047=0.2068
Torque_4 = F4 × 0 (zero because the lever arm is zero).

Add up those four torques to get "".

0.858-0.44-0.2068

=0.2112

you can find that the moment of inertia of a solid cylinder is:

I = mR²/2

(where "m" is the cylinder's mass).

3.9*.112/2

=0.023595

Now

0.2112=0.023595*

=8.95

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