Rotational Mechanics

Rotational Mechanics AP Physics C: Mechanics

Enough with the particles… • Do you ever get tired of being treated like a particle? • We can not continue to lump all objects together and pretend that they undergo the same motion when acted upon by the same force… • We will now study the rotation of rigid bodies.

What do we already know? Objects rotate about their center of mass. For rotational motion it is useful to consider tangential and radial components instead of x and y Ideas of circular motion and centripetal force.

What is a rigid body? • An extended object whose size and shape do not change as it moves. • This size and shape can not be neglected when modeling its motion. • Objects that are held together by “massless rods” of molecular bonds. • Some objects can be modeled as rigid during parts of their motion.

Three types of motion of a rigid body:

There is an rotational analogy to every concept of linear motion. • We have looked at the basics when studying circular motion:

Tangential Components: Tangential Velocity Rotational Velocity vt r Tangential Acceleration Rotational Acceleration ω

Radial (centripetal) Component vt Radial Velocity ar Radial Acceleration ω

Rotational Kinematic Equations:

Sign Conventions: Counter-clockwise is positive direction for ω and vt. So positive α can be speeding up in the ccw direction or slowing down in the cw direction.

A Rotating Crankshaft A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time interval? This is a rotating rigid body with constant angular acceleration Imagine painting a dot on the crankshaft. IF the dot is at θ=0 and t=0, at a later time, the dot will be at:

A Rotating Crankshaft A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time interval?

The Center of Mass A 500g ball and a 2kg ball are connected by a massless 50cm long rod. Where is the center of mass? What is the speed of each ball if they rotate about the center of mass at 40rpm?

The Center of Mass A 500g ball and a 2kg ball are connected by a massless 50cm long rod. Where is the center of mass?

The Center of Mass A 500g ball and a 2kg ball are connected by a massless 50cm long rod. What is the speed of each ball if they rotate about the center of mass at 40rpm?

What can we measure? • What can we measure and analyze with a bike in order to further understand rotational mechanics?

Other Rotational Analogs? Rotational Mass? Rotational Force? Rotational Energy? Rotational Momentum?

Rotational Energy? • Why does the Sun rise in the morning? Why do magnets stick together? Because everybody says so. Everybody. • - Michael Scott

Rotational Energy?

Rotational Energy Why must there be such a thing? All of the atoms in the object are moving so they must have kinetic energy! Translational Kinetic Energy Can we use our analogies to find an expression for Rotational Kinetic Energy?

Rotational KE for a particle traveling in a circle:

Moment of Inertia • Has nothing to do with a moment in time. The word comes from the Latin momentum which means motion. • The rotational analog to mass. • Describes the distribution of mass relative to the axis of rotation. • Is different for each shape and orientation. • An object with a large mass is hard to accelerate, an object with a large moment of inertia is difficult to rotate.

Moment of Inertia (Rotational Laziness) • Inertia is the resistance to changes in motion • Moment of inertia is the resistance to changes in rotation.

Moment of Inertia Consider an irregular shape that is rotating: The object’s rotational energy is the sum of the kinetic energies of each particle Moment of Inertia can be calculated as the sum of the contributions from each particle in an object

Moment of Inertia Calculating moment of inertia can be very difficult for odd shapes. I for many shapes has been tabulated and printed in handbooks for scientists and engineers.

Note about the axis… If the rotation axis is not through the center of mass, then rotation may cause the center of mass to move up or down in a gravitational field. The gravitational potential energy of the object will change as it spins. With no friction in the axle or other dissipative forces, the mechanical energy can be described as:

The three masses, held together by lightweight plastic rods, rotate about an axle passing through the right–angle corner. At what angular velocity does the triangle have 100mJ of rotational energy? 150g 6cm ω axle 300g 250g 8cm

A 1m long, 0.2kg rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of the rod as it hits the wall? Conservation of Energy:

Calculating Moment of Inertia: Like finding inertia, we can not simply place the object on a scale to find its moment of inertia. We must go through the calculation. Recall: Moment of Inertia can be calculated as the sum of the contributions from each particle in an object as Δm approaches zero it can be replaced with the differential dm.

Calculating Moment of Inertia: Tips Break the object into elements that you will sum together. Do this in a way that keeps the same distance from the axis for all particles in each element. You will sum the elements over a range of distances so you must find an expression to substitute dm with a differential dx, dy, or dz. Densities are helpful but not necessarily the only way to solve: For a complex object made up of parts with known moments of inertia, sum the terms of each to find part to find the moment of inertia of the object:

Find the moment of inertia of a circular disk of radius R and mass M that rotates on an axis passing through its center.

The four T’s in the diagram are made from identical rods. Rank in order, from largest to smallest, the moments of inertia for rotation about each dashed line. Which has the most mass distributed farthest from the axis??? Ia>Id>Ib>Ic

Parallel-Axis Theorem We have been calculating the moment of inertia for rotational axes that run through the center of mass. This theorem helps us if we wish to use an off-center axis but know where it is in relation to a parallel, on-center axis.

Parallel-Axis Theorem Find the moment of inertia of a thin rod with mass M and length L about an axis 1/3rd of the length from one end. 1/3rd cm d=L/6 Moment of inertia through the center of mass of a thin rod from table.

Rotational Force? Why must there be such a thing? Because all net forces do not cause rotation! How can rotational force be maximized? Apply a force at the proper location. How do we calculate it?

Torque • Torque is to rotational motion as force is to linear motion. • Torque is given the Greek symbol capital tau (τ)

Torque • Is greater with a greater force • Is greater if the force is applied farther from the axis or rotation. • Is greater if the angle of application of the force is perpendicular to the radial line.

Torque

Torque and Lever arm or Moment Arm

How much Torque does Luis Provide?

Rank the Torques τe>τa=τd>τb>τc

Analog to Newton’s 2nd Law No Change in motion (no acceleration) No Change in rotation EQUILIBRIUM

Torque due to gravity Center of mass is relative to the axis of rotation

Torque due to gravity The gravitational torque is found by treating the object as if all its mass were concentrated at the center of mass.

What is the torque on the 500kg steel beam?

Torque

Rotational Mechanics