Rotational Dynamics

Rotational Inertia (moment of inertia) Remember that inertia is the resistance an object has to movement, it depends solely on mass Rotational inertia measures the amount of torque it takes to get an object rotating, in other words it is the resistance of an object to accelerate angularly It depends not only on the mass of the object, but where the mass is relative to the hinge or axis of rotation The rotational inertia is bigger if more mass is located farther from the axis. Moment of Inertia variable is I In general I=mr2 (units are kg m2)

example If these cylinders have the same mass, which will reach the bottom of the ramp 1st? The solid one! It has less moment of inertia because its mass is evenly distributed and the hollow one has it mass distributed farther away from its rotational axis

Another example Which of these two rods will be harder to pick up? To spin? They will be the same to pick up because their mass is the same, but the one on the left will be harder to spin because its mass is located farther from axis of rotation.

Moments of Inertia – if needed formulas will be given bicycle rim filled can of coke baton baseball bat basketball boulder

How fast does it spin? For a given amount of torque applied to an object, its rotational inertia determines its rotational acceleration  the smaller the rotational inertia, the bigger the rotational acceleration

Same torque, different rotational inertia spins slow spins fast Big rotational inertia Small rotational inertia spins slow spins fast

Example Treat the spindle as a solid cylinder. What is the moment of Inertia of the spindle? b) If the tension in the string is 10N, what is the angular acceleration of the wheel? c) What is the acceleration of the bucket? What is the mass of the bucket? How far has the bucket dropped after 2.5 sec? a) 0.9 kgm2 b) 6.7 rad/s2 c) 4 m/s2 d) 1.7 kg e) 12.5 m

A 4 m beam with a 30 kg mass is free to rotate on a hinge A 4 m beam with a 30 kg mass is free to rotate on a hinge. It is attached to a wall with a horizontal cable. The cable is then cut, find the initial angular acceleration of the beam. mg θ Fx Fy 90o-θ θ + . θ = 35o

Extra physics c stuff

Continuous Masses Where do the equations for moment of inertia come from? Calculus is needed. This suggests that we will take small discrete amounts of mass and add them up over a set of limits. Consider a solid rod rotating about its CM.

Linear mass density Linear mass density = M/L dm= dl dm = M/L dl

The rod dr The CM acts as the origin in the case of determining the limits.

Parallel Axis Theorem This theorem will allow us to calculate the moment of inertia of any rotating body around any axis, provided we know the moment of inertia about the center of mass.

What if the rod were rotating on one of its ENDS? dr

The solid disk R r dx 2pr dx

The bottom line.. Will you be asked to derive the moment of inertia of an object? Possibly! Fortunately, most of the time the moment of inertia is given within the free response question.

A little trickier example Calculate the acceleration of the system when the pulley’s mass is not neglible: Assume m1 is more massive than m2 What you have to understand is that when the PULLEY is massive you cannot assume the tension is the same on both sides. Let’s first look at the F.B.D.s for both the pulley and the hanging masses. T1 FN T2 T2 T1 m1g m2g mpg

Example cont’ T1 T2 m2g m1g FN T2 T1 mpg

Example