Rotational Dynamics Moment of Inertia.

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1 Lecture 18 Chapter 10 Physics otational Dynamics Moment of nertia. Course website: Lecture Capture: , Fall 013, Lecture 18

2 Outline Chapter 10 Moment of nertia Parallel Axis Theorem otational kinetic energy olling , Fall 013, Lecture 18

3 Newton s nd law of rotation Force causes linear acceleration: (N. nd law): F ma Torque causes angular acceleration: is the Moment of nertia (rotational equivalent of mass) , Fall 013, Lecture 18

4 Moment of inertia of a single particle A point mass is located at a distance from an axis of rotation. A force is applied perpendicular to. Let s find a relation between torque and angular acceleration: 90 F N. nd law: m By definition: ecall, last class: As a result, torque is: ( m) FSin F F ma m a ( m ) Moment of inertia of a single particle: otational N. nd law: m , Fall 013, Lecture 18

5 Moment of inertia of many particle f we have many point masses m i, located at distances i from an axis of rotation. A force is applied perpendicular to. m m 3 m Moment of inertia of N masses: m 1 1 m m33 i1 1 m otational N. nd law: 1 N N m i ( m i i ) i1 i , Fall 013, Lecture 18

6 Example: Moments of nertia of two points Two point masses connected to a massless rod m 1 1 m 5kg(m) 7kg(m) 48kg m m 1 1 m 5kg(0.5m) 7kg(4.5m) 143kg m , Fall 013, Lecture 18 The distribution of mass matters

7 Moment of inertia for extended objects dm i mi The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation compare (f) and (g), for example , Fall 013, Lecture 18

8 Example: pulley and mass/physics An object of mass m is hung from a cylindrical pulley of radius and mass M and released from rest. What is the acceleration of the object? We have two objects in the system: - translational motion of m, described by N. nd law (1) F ma - rotational motion of M, described by the rotational N. nd law () - and there is a useful eq-n, which links these eq-ns: (1) (3) a tan a mg F T ma ( ) Sin90 (3) F ma a tan a F T a 1 y Physics is over. Now, it is pure Algebra. 3 eq-ns and 3 unknowns ( see table10 0) M F T F T mg , Fall 013, Lecture 18

9 Example: pulley and mass/algebra mg F T mg a F T F T (1) mg F T ( ) (3) ma a mg m F T F T a ma Or better ( see table 10 0) since ma Multiply both sides of (1) by and add (1) and () up: (F T disappears) a 1 M a mg m a y mg M m M F T F T mg , Fall 013, Lecture 18

10 Torque due to gravity We often encounter systems in which there is a torque exerted by gravity. The torque on a body about any axis of rotation is WSin MgSin The proof W Mg , Fall 013, Lecture 18

Example Problem: Falling rod What is the angular acceleration of the rod shown below, if it is released from rest, at the moment it is released? What is the linear acceleration of the tip?

11 Example Problem: Falling rod What is the angular acceleration of the rod shown below, if it is released from rest, at the moment it is released? What is the linear acceleration of the tip? otational motion of the rod is described by the rotational N. nd law Torque due to gravity (previous slide): WSin Mg Sin90 l Mg Mgl 1 Ml 3 a tan , Fall 013, Lecture 18 MgSin Mg l Mgl ( 1 Ml 3 ) 3g l l( 3g l) Each point on a rotating rigid body has the same angular acceleration (previous class)! So we can apply it to the tip!! 3g

Parallel Axis Theorem (w/out proof) The moment of inertia about any axis parallel to that axis through the center of mass is given by Mh h : moment of inertia about any parallel axis : moment of

12 Parallel Axis Theorem (w/out proof) The moment of inertia about any axis parallel to that axis through the center of mass is given by Mh h : moment of inertia about any parallel axis : moment of inertia about an axis through its center of mass M : total mass h : distance from a parallel axis to the center of mass. BTW: The moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space , Fall 013, Lecture 18

Parallel Axis Theorem: Example/Sphere Moment of inertia for the sphere, rotating about an axis through its center of mass Mr Moment of inertia for the sphere about an axis going through the edge of

13 Parallel Axis Theorem: Example/Sphere Moment of inertia for the sphere, rotating about an axis through its center of mass Mr Moment of inertia for the sphere about an axis going through the edge of the sphere? Apply Parallel Axis Theorem: Mh For a uniform sphere of radius r 0 Mr Mr Mr 0 Through an axis a distance >>r 0 from the center? Very small Mh 5 Mr0 M M So, in this case we got a Moment of inertia of a single particle , Fall 013, Lecture 18 >>r 0

14 v r i i m i otational Kinetic Energy Simple derivation: for pure rotation K K rot rot 1 1 m i v i 1 Krot mi ( ri ) 1 m i r i m i r i Since m i r i vi r i Ktrans 1 mv therefore , Fall 013, Lecture 18 K rot 1 Similar to linear motion Krot 1

15 Total kinetic energy olling motion can be resolved into two motions olling Translational otational = + Total Kinetic Energy K K rot K f there is no slipping tot 1 1 Mv v K tot 1 1 M 1 Ktot ( M ) , Fall 013, Lecture 18

16 ConcepTest 1 A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). n which case does the dumbbell acquire the greater center-of-mass speed? Dumbbell A) case (a) B) case (b) C) no difference D) it depends on the rotational inertia of the dumbbell Because the same force acts for the same time interval in both cases, the change in momentum must be the same, thus the velocity must be the same.

17 ConcepTest A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). n which case does the dumbbell acquire the greater energy? Dumbbell A) case (a) B) case (b) C) no difference D) it depends on the rotational inertia of the dumbbell f the velocities are the same, the translational kinetic energies must be the same. Because dumbbell (b) is also rotating, it has rotational kinetic energy in addition.

18 Sphere rolling down an incline , Fall 013, Lecture 18

19 Thank you See you on Wednesday , Fall 013, Lecture 18

20 Moment of inertia The distribution of mass matters here these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation. object1 object M 1 M , Fall 013, Lecture 18

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013 PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be