Ph i 218Phyysics 218 Chap 14 & 15people.physics.tamu.edu/mahapatra/teaching/ch14_15.pdfPhysics 218,...

Ph i 218Physics 218yChap 14 & 15Chap 14 & 15Prof. Rupak Mahapatra

Physics 218, Chapter 15 & 16

1

Angular Quantitiesg Q• Position Angle θ• Velocity Angular Velocity ω• Acceleration Angular Acceleration αAcceleration Angular Acceleration αMoving forward:

– Force– MassMass– Momentum– Energy

Physics 218, Chapter 15 & 16 2

Torqueq• Torque is the analogue of Force• Take into account the perpendicular

distance from axis– Same force further from the axis leads to more Torqueto more Torque


Slamming a doorgWe know this from experience:We know this from experience–If we want to slam a door really hard, we grab it at the endend

–If we try to push in the If we try to push in the middle, we aren’t able to

k l l h dmake it slam nearly as hard


Torque Continuedq

• What if we What if we change the angle at which angle at which the Force is

li d?applied?• What is the What is the “Effective Radius?”Radius?


Slamming a doorgWe also know this from experience:If t t l d – If we want to slam a door really hard, we grab it at the y , gend and “throw” perpendicular to the hingesto the hinges

– If we try to pushing towards y p gthe hinges, the door won’t even close


even close

Torqueq• Torque is our “slamming” ability• Need some new math to do Torque

Write Torque as τWrite Torque as τ

sin|F||r||| = θτ

Fr||||||

rrr×=τ

• To find the direction of the torque, wrap

Fr ×=τTo find the direction of the torque, wrap your fingers in the direction the torque makes the object twist


Vector Cross Productrrr

ΘSinB ACB A C

=

×=r

ΘSinB AC =

This is the last way of multiplying vectors we p y gwill see

•Direction from the “ i ht h d l ”“right-hand rule”

•Swing from A into B!


Vector Cross Product Cont…Multiply out, but use the p ySinθ to give the magnitude and RHR to magnitude, and RHR to give the directionˆˆ

)1( i kji

)0(sin 0ii ==× θ

)1( i jki

)1(sin kji ==×

θ

θ

)1(sin jki =−=× θ


Cross Product Examplep

ˆˆrj A i A A YX

r+=

j B i B Br

+=

i BA i Whj B i B B YX

rr+=

using BA is What ×

notation? Vector Unit Physics 218, Chapter 15 & 16 10

Torque and ForceqTorque problems are like Force q p

problems1 D f di1. Draw a force diagram2 Then sum up all the torques 2. Then, sum up all the torques

to find the total torque

Is t rque vect r?Is torque a vector?Physics 218, Chapter 15 & 16 11

Example: Composite Wheelp pTwo forces, F1 and

F F2Θ2F2, act on different radii of a wheel R and R

F2Θ2

wheel, R1 and R2, at different angles Θ1 and Θ2 Θ1 is a Θ1 and Θ2. Θ1 is a right angle.

If the axis is fixed If the axis is fixed, what is the net torque on the

Θ1torque on the wheel?

F1Physics 218, Chapter 15 & 16 12

1

Angular Quantitiesg Q• Position Angle θ• Velocity Angular Velocity ω• Acceleration Angular Acceleration αAcceleration Angular Acceleration αMoving forward:

– Force Torque τ– MassMass– Momentum– Energy


Analogue of MassgThe analogue of g fMass is called Moment of Moment of Inertia

Example: A ball of mass m moving in a gcircle of radius Raround a point has a pmoment of inertiaF=ma τ=Ια


F=ma τ=Ια

Calculate Moment of Inertia

Calculate the Calculate the moment of inertia for a ball f mass ball of mass m relative to m relative to the center of the circle R


Moment of Inertia•To find the mass of an object, just add up all the li l i f little pieces of massT fi d th m m t f To find the moment of inertia around a point just inertia around a point, just add up all the little moments

∫∑ == dmrI or mrI 22

p


Torque and Moment of Inertiaq

• Force vs. Torque Force vs. Torque F=ma τ = Iα

• Mass vs. Moment of Inertia

∑=⇒ or mrIm 2

∫= dmrI 2


Pulley and BucketyA heavy pulley, with y p yradius R, and known moment of inertia Imoment of inertia Istarts at rest. We attach it to a bucket attach it to a bucket with mass m. The f i ti t i friction torque is τfric.

Find the angular F gacceleration α


Spherical Heavy Pulleyp y yA heavy pulley, with Rradius R, starts at rest. We pull on an

R

attached rope with a constant force FT. It accelerates to an angular speed of ω in time t.

What is the moment of m m finertia of the pulley?


Less Spherical Heavy Pulleyp y yA heavy pulley, with radius

R W ll RR, starts at rest. We pull on an attached rope with constant force FT It

Rconstant force FT. It accelerates to final angular speed ω in time t.

A better estimate takes into account that there is friction in the system This friction in the system. This gives a torque (due to the axel) we’ll call this τfric.fric

What is this better estimate of the moment of Inertia?


Example of Cross ProductpThe location of a body zis length r from the

origin and at an z

angle θ from the x-axis. A force F acts on the body purely in the y direction.

What is the Torque on the body?

yθy

xθ


Calculate Moment of Inertia1.Calculate the

moment of inertia for a ball of mass for a ball of mass m relative to the center of the center of the circle R

2.What about lots of points? For f p Fexample a wheel


Rotating RodgA uniform rod of mass m, length l, and moment of

inertia I = ml2/3 rotates around a pivot It is inertia I = ml /3 rotates around a pivot. It is held horizontally and released.

Find the angular acceleration α and the linear Find the angular acceleration α and the linear acceleration a at the end. Where, along the rod, is a = g?g


Two weights on a bargFind the middled t emoment of i tiinertia for thefor the two different A

Physics 218, Chapter 15 & 16 24Axes

Moments of Inertia


Ph i 218Phyysics 218 Chap 14 & 15people.physics.tamu.edu/mahapatra/teaching/ch14_15.pdfPhysics 218,...

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