Circular motion

Lecture #6, by I. Bars

In this lecture we discuss rotational inertia, torque, rotational velocity, rotational acceleration and angular momentum. Similar concepts occur in planetary motion (which are close to but not exactly circular) as we will see in our discussion of gravity.
Credit for grpahics, CD-Library, Hewitt.

Circular motion, note position, velocity and acceleration vectors.
Observe also the motion of the pendulum.

Angular speed versus linear speed. The farther you are from the center the faster the linear speed (IF the angular speed is the same).

See video "Rotational Speed" (coins on rorating phonograph disk)
See also the pendulum motion short versus long string.

v = r w

Linear speed (v, units m/s), angular speed (w , units radians/s , 2pi radians = 360 degrees).
Which horse moves faster in a merry-go-round - a horse near the outside rail or a horse near the inside rail?
For a given angular speed, the linear distance travelled on circles of different sizes is different (d=vt=r w t). Design of railroad train wheels !!! (see Next Time Questions #10)

Rotational Inertia. See pendulum, Next Time Questions #1, videos on "Rotational Inertia using weighted pipes" , "... using a hammer".

The greater the distance of mass concentration, the greater the inertia (resistance to rotation).

I=Mr²(point mass)

^{See Fig. 8.14 for rotational inertias of different
objects rotating about different centers}

Center of mass. See video "Locating the Center of Mass".

where is your center of mass? Why does a hiker with a heavy backpack lean forward?
Area of support, toppling, stability. See Next Time Questions #2, see video "Toppling". Leaning Pisa tower. Stability of lever arm.

Torque. Why a ball rolls down a hill as opposed to slide down? Why the horizontal pencil rotates?

See videos "Difference between force and weight", "Why a ball rolls downhill"

Solid disk versus ring.
Same size and mass.
Which one rolls down faster?
Why?

I=Mr²(ring)
I=Mr²/2(disk)

Torque = Rotational inertia x Rotational acceleration

To maintain rotational motion must have a force toward the center, and therefore an acceleration toward the center.

See Circular motion , or the pendulum.
see video "Centripetal Force" (water bucket).
There is an acceleration toward the center since a particle undergoing rotational motion must change direction to stay on the circle. This centripetal acceleration is given by a=v²/r =w²r.
The centripetal force is illustrated by a swinging stone attached to a string. It balances the centrifugal force.
Question: Observe a satellite around the earth at different velocities. What produces the centripetal force?
Simulated gravity in a space station. See video "Simulated Gravity"

Angular momentum. Angular momentum is conserved for isolated systems (angular momentum at any time is the same, before an event or after an event). This is a similar rule to conservation of linear momentum.

L = I w

See video "Conservation of angular momentum" using a rotational platform. How do you explain the change in the rotational speed?

Reading assignment: Read Ch.8, Hewitt. Study the answers to the Next Time Questions.
Homework #1, part f : Quiz in Ch.8

Back to homepage