Rotational Energy

1. Moment of Inertia and Rotational Energy: K_rot = .5 Iv²
Application: Throwing rope over a limb.

Discussion: relationship between s,r & u, v & v, a & a

2. Rotational Energy Problem: Does the object rotate about its center of mass or about the edge? (If not, the cm would rotate about the pivot point.) Supposed to do the demo with two jars of liquid, one thin, one thick.

2. Torque & Center of mass.

Torque = the pain in the wrist due to the twisting force.
Officially, t = r x F, it's direction is obtained by the right hand rule. Point fingers of right hand in direction of r so that can curl fingers in direction of F, and the thumb of the right hand will point in the direction of t . It's magnitude is t = rF sin u, where u is the angle between r & F, but be careful how you choose this angle. CCW torque is positive, and CW torque is negative. You must select the angle properly to get the correct sign. Extend r, and rotate ccw to F to get u.

Do the demo with finding cm for the meterstick with wts attached and discuss the point: Torque on the whole system behaves as if all its mass was located at it's center.
(Show them the box pictures).

Now find the cm of a 2 diminsional figure.(Demo the plywood triangle first, then do problem in the xy plane).

Next explain the Gyro trick, t = DL/Dt (relate to F = Dp/Dt ), so precesses in the direction that makes DL be in the direction of t .

The Disk? conservation of energy, do the before and after energy chart.

Write an expression for the rotational inertia of each group of masses shown below.

About the axis through the center and perpendicular to the page.	About the axis through the center and perpendicular to the page.
Which ring is easiest to start (or stop) rotating? Explain.		About which axis is it easiest to (start or stop) the rotation of the above masses? Explain.

DETERMINING TORQUES: THE CROSS PRODUCT METHOD

The second method for determining torques is more suitable for situations in which forces are exerted on beams, ladders, crates, and so forth at angles other than 90° . The method is illustrated below and summarized on the facing page.

EXAMPLE 6.4 Determine the torque on
the beam caused by the rope tension T₁ .

SOLUTION
* Construct a separate coordinate system
and draw a radial vector r from the origin to
the position where the force acts on the beam.
Extend this vector with a dotted line beyond r.
* Then, draw a line representing the force
whose torque you wish to determine. The
tail of the force vector should be at the tip
of the radial vector.
* Next, draw a curved, dashed line in the
countcrclockwise direction from the extension
of r to the force vector. This curved arc indicates
1 the angles (between 0 and 360°) used to
determine the magnitude of the torque.
* The magnitude of the torque is the product
of the magnitude of r, the magnitude of the force
T1 and the sine of the angle ¢.
* Direction of torque: Officially, the expression
for the torque is the cross product of r and the force, T₁:

T = r x T₁

The torque points along a line perpendicular to t
he plane that contains r and T₁ and has the magnitude
given at the right. The torque is positive (out of the
plane) for counterclockwise torques and negative
(into the plane) for clockwise torques. The value
of sin ø determines the sign of the torque.

This torque will have a negative sign because the angle is greater than 180° and less than 360°. It tends to rotate the beam in the clockwise direction.

For each situation below, determine the torque caused by force F about the origin of the coordinate system shown in the sketch. The magnitude of F is 100 N.

Does your answer have the correct sign?	Does your answer have the correct sign?	Does your answer have the correct sign?
Answer:	Answer:	Answer:

Construct a free-body diagram and determine the net torque acting on the objects below. Assume that g = 10 m/s².

Person pushing a 100-Kg crate with 400N horizontal force. The center of gravity is at the center of the crate.	The system is stationary.
FBD	FBD
Is the crate rotationally stable?

For each situation shown below, determine the direction of the angular velocity, the direction of the angular acceleration, and the directions of the resultant torque. Place these directions (in = into the paper, out = out of the paper, or 0 = zero) in the table.

Initially rotating ccw

Initially rotating cw

Initially rotating cw

Initially rotating ccw

(b.) Describe in words how the angular velocity changes.

I.

II.

III.

IV.

(c.) Complete the table.

(d.) Bases on the information in the table, is …t propartional to v? Explain. (e.) Is …t proportional to a?

Four 60-kg people stand 3.0 m from the center
of a freely rotating disc. Initially the angular
velocity of the disc is +1.0 rads/s (ccw). The
people then move inward so that they are 1.0m
from the center of the disc. Now what is the angular
velocity of the disc? The rotational inertia of the
disc is 160 kg * m2.

(a) Choose a system. Is its angular momentum conserved? Explain.

(b) Does the rotational inertia of the system increase or decrease? Explain.

(c) Does the angular velocity of the disc increase or decrease? Explain.

(d) Determine the initial and final rotational inertia of the system.

I₀ = I =

(e) Apply the conservation of angular momentum principle to this problem.

(f) Solve for the unknown final angular velocity.

(g) Evaluation:

• Are units correct?
  
  
• Is the magnitude reasonable?
  
  
• Does the answer agree with qualitative analysis in (b) and (c) ? Neutron Star
  

As a star nears the end of its life and begins to cool, its mass collapses because of the pull of gravity. The protons and elctrons combine to form neutrons. Our sun of mass 2.0 x 10³⁰ kg and initial 1.4 x 10⁸ m now rotates once about its axix each month. If many years from now it became a neuron star of radius of 10 km, what now would be the time needed for one rotation about its axis. (A teaspoon of neutron star weighs nearly a billion tons!)


List KNOWN INFORMATION:

INITIAL FINAL
_____________________________

Identify UNKNOWN:

(b) Is angular momentum conserved during the process? Explain.

(c) Does the rotational inertia of the sun increase or decrease as it collapses? Explain.

(d) Does the angular velocity increase or decrease? Explain.

(e) Does the time interval for one rotation increase or decrease? Explain.

(f) Apply the conservation of angular momentum principle to this process.

(g) Write an equation that relates the angular velocity and the time interval for one rotation (called the period T).

(h) Solve for the desired unknown.

COMMENT : Pulsars that emit radiation with the period of 0.03 to 4s are thought to be rotating neutron stars.

ALPS Kit, Box 4013, University Pk, v-24A FIPSE Project Las Cruces, NM 88003 (Rotational Motion and Statics)