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Chapter 9
Rotational Dynamics
9.1 The Action of Forces and Torques on Rigid Objects
In pure translational motion, all points on anobject travel on parallel paths.
The most general motion is a combination oftranslation and rotation.
9.1 The Action of Forces and Torques on Rigid Objects
According to Newton’s second law, a net force causes anobject to have an acceleration.
What causes an object to have an angular acceleration?
TORQUE
9.1 The Action of Forces and Torques on Rigid Objects
The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.
9.1 The Action of Forces and Torques on Rigid Objects
DEFINITION OF TORQUE
Magnitude of Torque = (Magnitude of the force) x (Lever arm)
FDirection: The torque is positive when the force tends to produce a counterclockwise rotation about the axis.
SI Unit of Torque: newton x meter (N·m)
9.2 Rigid Objects in Equilibrium
If a rigid body is in equilibrium, neither its translational motion nor its rotational motion changes.
0 xF 0yF 0
0 yx aa 0
9.2 Rigid Objects in Equilibrium
EQUILIBRIUM OF A RIGID BODY
A rigid body is in equilibrium if it has zero translationalacceleration and zero angular acceleration. In equilibrium,the sum of the externally applied forces is zero, and thesum of the externally applied torques is zero.
0 0yF0 xF
9.2 Rigid Objects in Equilibrium
Example 3 A Diving Board
A woman whose weight is 530 N is poised at the right end of a diving boardwith length 3.90 m. The board hasnegligible weight and is supported bya fulcrum 1.40 m away from the leftend.
Find the forces that the bolt and the fulcrum exert on the board.
9.2 Rigid Objects in Equilibrium
022 WWF
N 1480
m 1.40
m 90.3N 5302 F
22
WWF
9.2 Rigid Objects in Equilibrium
021 WFFFy
0N 530N 14801 F
N 9501 F
9.3 Center of Gravity
DEFINITION OF CENTER OF GRAVITY
The center of gravity of a rigid body is the point at whichits weight can be considered to act when the torque dueto the weight is being calculated.
9.3 Center of Gravity
21
2211
WW
xWxWxcg
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
TT maF
raT rFT
2mr
Moment of Inertia, I
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
2
2222
2111
NNN rm
rm
rm
2mr
Net externaltorque Moment of
inertia
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FORA RIGID BODY ROTATING ABOUT A FIXED AXIS
onaccelerati
Angular
inertia
ofMoment torqueexternalNet
I
2mrIRequirement: Angular acceleration must be expressed in radians/s2.
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
Example 9 The Moment of Inertia Depends on Wherethe Axis Is.
Two particles each have mass and are fixed at theends of a thin rigid rod. The length of the rod is L.Find the moment of inertia when this object rotates relative to an axis that is perpendicular to the rod at(a) one end and (b) the center.
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
(a) 22222
211
2 0 LmmrmrmmrI
Lrr 21 0mmm 21
2mLI
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
(b) 22222
211
2 22 LmLmrmrmmrI
22 21 LrLr mmm 21
221 mLI
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
9.5 Rotational Work and Energy
FrFsW
rs
Fr
W
9.5 Rotational Work and Energy
DEFINITION OF ROTATIONAL WORK
The rotational work done by a constant torque in turning an object through an angle is
RW
Requirement: The angle mustbe expressed in radians.
SI Unit of Rotational Work: joule (J)
9.5 Rotational Work and Energy
22122
2122
21 ImrmrKE
22212
21 mrmvKE T
rvT
9.5 Rotational Work and Energy
221 IKER
DEFINITION OF ROTATIONAL KINETIC ENERGY
The rotational kinetic energy of a rigid rotating object is
Requirement: The angular speed mustbe expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)
9.6 Angular Momentum
DEFINITION OF ANGULAR MOMENTUM
The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia and its angular velocity with respect to thataxis:
IL
Requirement: The angular speed mustbe expressed in rad/s.
SI Unit of Angular Momentum: kg·m2/s
9.6 Angular Momentum
PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM
The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.
9.6 Angular Momentum
Conceptual Example 14 A Spinning Skater
An ice skater is spinning with botharms and a leg outstretched. Shepulls her arms and leg inward andher spinning motion changesdramatically.
Use the principle of conservationof angular momentum to explainhow and why her spinning motionchanges.
Problems to be solved
• 9.6, 9.12, 9.14, 9.21, 9.25, 9.40, 9.51, 9.61, 9.69, 9.74
• B9.1 A bicycle wheel has a mass of 2kg and a radius of 0.35m. What is its moment of inertia? Ans: 0.245kgm2
• B9.2 A grinding wheel, a disk of uniform thickness, has a radius of 0.08m and a mass of 2kg. (a) What is its moment of inertia? (b) How large a torque is needed to accelerate it from rest to 120rad/s in 8s? Ans: (a) 0.0064kgm2 (b) 0.096Nm
• B9.3 A student holding a rod by the centre subjects it to a torque of 1.4Nm about an axis perpendicular to the rod, turning it through 1.3rad in 0.75s. When the student holds the rod by the end applies the same torque to the rod, through how many radians will the rod turn in 1.0s?
Ans: 0.58rad
• B9.4 An ice skater starts spinning at a rate of 1.5rev/s with arms extended. She then pulls her arms in close to her body, resulting in a decrease of her moment of inertia to three quarters of the initial value. What is the skater’s final angular velocity? Ans: 2rev/s