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Wize AP Physics C: Mechanics Textbook > Unit 5: Rotation (14-20%)

Rotational Analog of Newton's Second Law

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Rotational Analog of Newton's Second Law


Torque is the rotational analog for force. When we apply a torque, we are changing the angular acceleration (making it spin faster or slower). This change is given by Newton's second law for rotations

∑τ =I⋅α\boxed{\sum_{ }^{ }\tau\ =I\cdot\alpha}

where IIis the moment of inertia and α\alphais the angular acceleration.


Exam Tip
Above Law looks a lot like Newton's second law, except we use moment of inertia for mass, and angular acceleration for acceleration.

Wize Tip
In the above formula, α\alpha is in rad/s2.




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Example: Rotating Rod


A thin uniform rod (length=1m, mass = 5kg) is pivoted about a horizontal, frictionless pin through one end of the rod. The rod is released when it makes an angle of 60°60\degree above the horizontal. What is the angular acceleration of the rod at the instant it is released? (The moment of inertia of a rod with length of L and mass of m about one of its ends is I=13mL2I=\frac{1}{3}mL^2)


Solution:

The only force acting on the rod is gravity, and so this produces a torque at the centre of mass. We write: ∑τ=Iα\sum_{ }^{ }{\tau}=I\alpha

To find the torque caused by gravity, we need the distance from the pivot point to the center of mass which is half of the length of rod and also the angle gravity force makes with the position vector along the rod. This angle is 90−60=30°90-60=30\degree
(mg)L2sin⁡(30)=Iα=(13mL2)α(mg)\frac{L}{2}\sin(30)=I\alpha=\left(\frac{1}{3}mL^2\right)\alpha
Solving for α\alphawe get:


α=mg(sin⁡30) L213mL2=3gsin⁡302L=3(9.8)sin⁡302L=7.35 rad/s2\alpha=\frac{mg(\sin30)\ \frac{L}{2}}{\frac{1}{3}mL^2}=\frac{3g\sin30}{2L}=\frac{3\left(9.8\right)\sin30}{2L}=7.35\ rad/s^2




A 10 kg mass is hanging from a rope wrapped around a cylindrical pulley. The falling mass causes the pulley to rotate. The mass of the pulley is 3 kg. (The moment of inertia of a solid cylinder with radius R and mass of M about its axis of symmetry is I=12MR2I=\frac{1}{2}MR^2)

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What is the acceleration of the falling mass? (choices are in m/s2m/s^2)