PHYSC 1114, Lecture 17 Contents
PHYSC 1114, Lecture 17 Contents: 1◦ Assigned Homework 9: Chapter 6, Problem 26. Chapter 8, Problems 26, 30, 34, 36, 44, 50, 58. This is due Wednesday in class. 2◦ Rotational Dynamics and Equilibrium: a. Discussed Extended Second Law of Newton, Work, Kinetic Energy, Angular Momentum, see sections 8-7 through 8-9 in book. b. Discussed Equilibrium, see sections 9-1 through 9-3. c. Showed two movies and performed one experiment on Center of Percussion. d. Showed and discussed transparencies of Figures 8-17, 8-20, 8-24, 8-28. 3◦ Extended Second Law of Newton: An extended body can have a motion that is the sum of atranslational motion of its center of mass, governed by F = ma, and of a rotational motion, described by τ = Iα. When there is no net force and no net torque, this reduces to the Extended First Law of Newton: If no net force and no net torque is exerted on a rigid body, the body is at rest or continues in a superposition of uniform translation and rotation. This will be used in the next chapter. There is also an Extended Third Law of Newton, namely Action = −Reaction for the forces and torques which two bodies exert on each other. i
3◦ Rotational Work: W = F ∆l = F r∆θ = τ ∆θ WRot = τ ∆θ,
SI Unit: J = N m
4◦ Rotational Kinetic Energy: KE =
1
2 mv
2
=
1
2
2 m(rω)
KERot = 12 Iω 2 ,
=
1 2
mr2 ω 2 = 12 Iω 2
KETransl = 12 mv 2
E = PE + KE = mgy + 12 mv 2 + 12 Iω 2 The kinetic energy has a translational and a rotational contribution. 5◦ Example 8-14 in the book was treated. 6◦ Angular Momentum: Using α = ∆ω/∆t, we rewrite Newton’s Second Law for rotational motion, τ = Iα, as
τ =I
∆ω ∆L = , ∆t ∆t
where
SI Unit: kg m2 /s [This is the rotational analogue of F = ∆p/∆t.] L = Iω,
7◦ Conservation of Angular Momentum: When there is no net torque acting on a rotating body, then the angular momentum L = constant. ii
Remark: As is taught in more advanced physics courses, the set of fundamental conservation laws expresses the fact that different observers have the same laws of physics: The Law of Conservation of Energy says that physics is the same at all times. The Law of Conservation of Linear Momentum says that physics is the same at all places. The Law of Conservation of Angular Momentum says that physics is the same for observers that are rotated with respect to one another. 8◦ Equilibrium: The First Law of Newton requires the three Equilibrium Conditions for the Rigid Body:
Fx = 0,
Fy = 0,
τ =0
[Strictly spoken, we also need Fz = 0 and τ = 0 about axes that are not perpendicular to the xy-plane, but that is beyond this course.] 9◦ Couple: For bodies that are rotating about a fixed axis, the sum of the forces is zero. Any applied force is countered by a reaction force by the axis. If there is no fixed axis, the rotational state of the body can be changed by a couple, a pair of equal but opposite forces of magnitude F , separated by an arm of length l. This couple provides a torque of magnitude F l. 10◦ Examples 9-4 and 9-6 in the book were treated. iii
F
l
F
11◦ Movies and Experiments The Center of Percussion is the rotational analogue of the Center of Mass. It will not appear on any of the exams. i. Movie: Free-Fall Paradox: Falling Chimney An inclined plane hinged at one end (see figure) can be used to study angular acceleration. A small cup is attached near the end and a ball is placed at the very end. The incline is raised to an angle of 35◦ and dropped. The ball is seen to leave the surface of the incline and then to fall into the cup! All of the mass of the incline can be considered to be concentrated in the center of percussion, two thirds from the axis of rotation. This is the rotational analogue of the center of mass. The center of percussion falls according to the acceleration of gravity g = 9.80 m/s2 , just like the ball. Points on the incline closer to the axis—like the center of mass which is in the middle of the incline—experience an acceleration downward less than g. But the cup falls with a higher acceleration and reaches the bottom before the ball, as it is further from the axis than the center of percussion. ii. Movie: Center of Percussion: Impulsive Forces Delivered to a Baseball Bat Baseball players know that unless the ball is hit at just the right spot on the bat a sharp sting is felt in the hands. This spot is called the center of percussion. If the ball is hit here, no vibration is felt in the handle. iv
To determine this special spot, the bat is suspended between two horizontal bars. Next to it a small mass is hung on a string, with length adjusted so that the bat and the mass swing back and forth with the same frequency, once put in motion. This determines the center of percussion which is then marked on the bat. An arrow is attached to the top of the bat and can rotate about a fixed axis somewhat higher (see figure). Applying an impulsive force by hitting the bat below the center of percussion makes the handle move to the right and the arrow point left. Hitting the bat above the center of percussion, for example near its center of mass, makes the arrow point right. The handle then has a translational acceleration to the left, which would be felt as a sharp jolt to the hands. When the bat is hit directly at the center of percussion, this makes the handle experience an angular tangential acceleration to the right and a translational acceleration to the left, which cancel each other. The batter would feel no stinging vibration in this case. iii. Experiment: Like in the righthand picture for Movie 2, a baseball bat is hanging down from its handle and it is hit at different heights with a stick. One can hear a difference in sound, with a more full sound if the bat is hit away from the Center of Percussion and a more dull sound if it is hit nearby. v
3◦ Rotational Work: W = F ∆l = F r∆θ = τ ∆θ WRot = τ ∆θ,
SI Unit: J = N m
4◦ Rotational Kinetic Energy: KE =
1
2 mv
2
=
1
2
2 m(rω)
KERot = 12 Iω 2 ,
=
1 2
mr2 ω 2 = 12 Iω 2
KETransl = 12 mv 2
E = PE + KE = mgy + 12 mv 2 + 12 Iω 2 The kinetic energy has a translational and a rotational contribution. 5◦ Example 8-14 in the book was treated. 6◦ Angular Momentum: Using α = ∆ω/∆t, we rewrite Newton’s Second Law for rotational motion, τ = Iα, as
τ =I
∆ω ∆L = , ∆t ∆t
where
SI Unit: kg m2 /s [This is the rotational analogue of F = ∆p/∆t.] L = Iω,
7◦ Conservation of Angular Momentum: When there is no net torque acting on a rotating body, then the angular momentum L = constant. ii
Remark: As is taught in more advanced physics courses, the set of fundamental conservation laws expresses the fact that different observers have the same laws of physics: The Law of Conservation of Energy says that physics is the same at all times. The Law of Conservation of Linear Momentum says that physics is the same at all places. The Law of Conservation of Angular Momentum says that physics is the same for observers that are rotated with respect to one another. 8◦ Equilibrium: The First Law of Newton requires the three Equilibrium Conditions for the Rigid Body:
Fx = 0,
Fy = 0,
τ =0
[Strictly spoken, we also need Fz = 0 and τ = 0 about axes that are not perpendicular to the xy-plane, but that is beyond this course.] 9◦ Couple: For bodies that are rotating about a fixed axis, the sum of the forces is zero. Any applied force is countered by a reaction force by the axis. If there is no fixed axis, the rotational state of the body can be changed by a couple, a pair of equal but opposite forces of magnitude F , separated by an arm of length l. This couple provides a torque of magnitude F l. 10◦ Examples 9-4 and 9-6 in the book were treated. iii
F
l
F
11◦ Movies and Experiments The Center of Percussion is the rotational analogue of the Center of Mass. It will not appear on any of the exams. i. Movie: Free-Fall Paradox: Falling Chimney An inclined plane hinged at one end (see figure) can be used to study angular acceleration. A small cup is attached near the end and a ball is placed at the very end. The incline is raised to an angle of 35◦ and dropped. The ball is seen to leave the surface of the incline and then to fall into the cup! All of the mass of the incline can be considered to be concentrated in the center of percussion, two thirds from the axis of rotation. This is the rotational analogue of the center of mass. The center of percussion falls according to the acceleration of gravity g = 9.80 m/s2 , just like the ball. Points on the incline closer to the axis—like the center of mass which is in the middle of the incline—experience an acceleration downward less than g. But the cup falls with a higher acceleration and reaches the bottom before the ball, as it is further from the axis than the center of percussion. ii. Movie: Center of Percussion: Impulsive Forces Delivered to a Baseball Bat Baseball players know that unless the ball is hit at just the right spot on the bat a sharp sting is felt in the hands. This spot is called the center of percussion. If the ball is hit here, no vibration is felt in the handle. iv
To determine this special spot, the bat is suspended between two horizontal bars. Next to it a small mass is hung on a string, with length adjusted so that the bat and the mass swing back and forth with the same frequency, once put in motion. This determines the center of percussion which is then marked on the bat. An arrow is attached to the top of the bat and can rotate about a fixed axis somewhat higher (see figure). Applying an impulsive force by hitting the bat below the center of percussion makes the handle move to the right and the arrow point left. Hitting the bat above the center of percussion, for example near its center of mass, makes the arrow point right. The handle then has a translational acceleration to the left, which would be felt as a sharp jolt to the hands. When the bat is hit directly at the center of percussion, this makes the handle experience an angular tangential acceleration to the right and a translational acceleration to the left, which cancel each other. The batter would feel no stinging vibration in this case. iii. Experiment: Like in the righthand picture for Movie 2, a baseball bat is hanging down from its handle and it is hit at different heights with a stick. One can hear a difference in sound, with a more full sound if the bat is hit away from the Center of Percussion and a more dull sound if it is hit nearby. v
