phys10-chap10-dynamicsofrotationalmotion

7/26/2019 Phys10-Chap10-DynamicsOfRotationalMotion

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-> measure of the tendency of a force to cause rotation

F r

torqueRadial distance fromaxis of rotation to

the point of action

sinrF

Magnitude of Torque

Angle between radius vector and fo

r

Note: Cross product operation obeys right hand rule.

F r

Torque and Rotation

Into the page torque-> clockwise rotation

(negative)

Out of the page torque-> counterclockwise

rotation (positive)

Example1. Applying a Torque

A weekend plumber, unable to loosen a pipe fitting, slips a piece of scrap pipe (a “chover his wrench handle. He then applies his full weight of 900 N to the end of the by standing on it. The distance from the center of the fitting to the point wheweight acts is 0.80m, and the wrench handle and cheater make an angle of 19o whorizontal. Find the magnitude and direction of the torque he applies about the c ethe pipe fitting



Rotational Analogue of Newton’s 2nd Law

5

I

Net (total)torque appliedon a rigid body

Note: The direction of the net torque is the same with the angular acceleration.

Example2. An unwinding cable

Find the accelerationof the cable.

-> combined translation (linear motion) and rotation

7

22

2

1

2

1

cmcm I mv K

Translation(linear)

Condition for Rolling w/out slipping

Rotation

r vcm

Rolling w/out Slipping

r vcm



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Example3. Speed of a primitiveyo-yo

A primitive yo-yo is made by wrapping a string several timesaround a solid cylinder with mass 0.100 kg and ra dius 0.05m.You hold the end of the string stationary while releasingthe cylinder with no initial motion. The string unwinds but does not slip or stretch as the cylinder drop or rotates.(a) Find the velocity of the center of mass of the yo-yo

after it has dropped a distance 1m.(b) Compare the velocity is just dropped directly with nostring (no rotation) from that height.

Example4. Race of rolling bodies

In a physics demonstration, an instructor “races” various round rigid bodies by relthem from rest at the top of an inclined plane. What shape should a body have to the bottom of the incline first?

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cm I

Valid only if:

1 . The axis of rotation is an axis of symmetry

2. Theaxis must not change direction

am F

translation

rotation

Example5. Accelerationof a primitiveyo-yo

For the primitive yo-yo in the previous example, find the downward acceleration atension in the string.



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Example6. Acceleration of a rolling sphere

A solid bowling ball rolls without slipping down the return ramp at the side of the alley.The ramp is inclined at angle =30o to the horizontal. What are the ball’s acceleration andmagnitude of frictional force on the ball? Treat the ball as a uniform solid sphere, ignorethe finger holes.

Work Done by a Constant Torque

W

Power Output by a Constant Torque

P

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Example7. Engine Power and torque

The power output of an automobile engine is advertised to be at 200hp at 6000 rpm.What is the corresponding torque?

Example8. Engine Power and torque

An electric motor exerts a constant torque of 10 Nm on a grindstone mounted on its shaft.The moment of inertia of the grindstone about the shaft is 2.0 kg m 2. If the systemstarts from rest, find the work done by the motor in 8.0 s and the kinetic energy at thistime. What was the average power delivered by the motor?

-> analogue of linear momentum

vmr

pr L

Magnitude of angular momentum

sinmvr L



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Angular momentuma nd momentof inertia

iiii r vm L

ii r v

(Singleparticle)

2

iii r m L

2

iir m L

I L

(systemof particles)

For any system of particles, the rate of change of angular momentum of a body eqthe net torque acting on it

t

L

CONSERVATION of ANGULAR MOMENTUM

f f ii

f i

I I

L L

If the net torque on a system is zero, then the angular momentumis constant (co

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Example9. Anyone can be a ballerina

An acrobatic physics professor stands atthe center of a turntable, holding his armsextended horizontally with a 5.0 kg dumbbellin each hand. He is set rotating about avertical axis, making 1 revolution in 2.0s.Find the prof’s new angular velocity if hepulls the dumbbells into his stomach. Hismoment of inertia (w/out the dumbbells) is3.0 kg m2 when his arm a re outsretched,dropping to 2.2 kg m2 when his hands are athis stomach. The dumbbells are 1.0m fromthe axis initially and 0.20m from it at theend.

Example10. R otational Collision

Suppose the flywheel A has a mass of 2.0 kg,a radius of 0.20m and an initial angular speedof 50 rad/s and that clutch plate B has amass 4.0 kg, a radius of 0.10 m and an initialangular speed of 200 rad/s. Find thecommon final angular speed after the disks are pushed into contact. What happens to thekinetic energy during this process?



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Example11. Angular momentum in a crime bust

A door 1.00 m wide, of mass 15 kg, is hinged atone side so that it can rotate without frictionabout a vertical axis. It is unlatched. A policeofficer fires a bullet with mass 10g and aspeed of 4m/s into the exact center of thedoor, in a direction perpendicular to the plane

of the door. Find the angular speed of thedoor just after the bullet embeds itself in thedoor. Is the kinetic energy conserved?

PRECESSION -> happens when the axis of a rotating rigid body changes direction

GYROSCOPE

-> anything that undergoes precession

(Ex. Earth)

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Prob 1. Calculate the net torque about point O for the twoforces applied. The rod and both forces are in the plane ofthe page.

Prob 2. A grindstone in the shape of a solid disk with diameter 0.520m, and a mass of 50.0 kg is rotating at 850rev/min. You press an ax against the rim with a force of160 N, and the grindstone comes to rest in 7.50 s. Find thecoefficient of friction between the ax and the grindstone.

Prob 3. A hollow, spherical shell w/ mass 2.00 kg rolls without slipping down a 38.0 o slopethe acceleration, the frictional force, and the minimum coefficient of friction needed to slipping.(b) How would your answers change if the mass were doubled?

Prob 5. Suppose that an asteroid travelling straight toward the center of the earth were with our planet at the equator and bury itself just below the surface. What would have tmass of this asteroid, in terms of the earth’s mass M, for the day to become 25.0% longepresently is as a result of this collision? Assume that the asteroid is very small compareearth and that the earth is uniformthroughout.

Prob 4. Find the magnitude of the angular momentum of the second hand on a clock abouthrough the center of the clock face. The clock hand has a length of 15.0 cm and mass 6.0the second hand to be a slender rod rotating w/ constant angular velocity about one end.

phys10-chap10-dynamicsofrotationalmotion

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