Ph i 218Physics 218yChap 14 & 15Chap 14 & 15Prof. Rupak Mahapatra
Physics 218, Chapter 15 & 16
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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
Physics 218, Chapter 15 & 16 3
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
Physics 218, Chapter 15 & 16 4
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?
Physics 218, Chapter 15 & 16 5
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
Physics 218, Chapter 15 & 16 6
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
Physics 218, Chapter 15 & 16 7
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!
Physics 218, Chapter 15 & 16 8
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
Physics 218, Chapter 15 & 16 9
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
Physics 218, Chapter 15 & 16 13
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
Physics 218, Chapter 15 & 16 14
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
Physics 218, Chapter 15 & 16 15
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
Physics 218, Chapter 15 & 16 16
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
Physics 218, Chapter 15 & 16 17
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
Physics 218, Chapter 15 & 16 18
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?
Physics 218, Chapter 15 & 16 19
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?
Physics 218, Chapter 15 & 16 20
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
Physics 218, Chapter 15 & 16 21
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
Physics 218, Chapter 15 & 16 22
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
Physics 218, Chapter 15 & 16 23
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
Physics 218, Chapter 15 & 16 25