Rotational Dynamics A wrench floats weightlessly in space. It is subjected to two forces of equal and opposite magnitude: Will the wrench accelerate? A. yes B. no C. kind of?
Rotational Dynamics 10.1-3 Rotational Variables 10.4-5 Moment of Inertia & Kinetic Energy 10.6 Torque 10.7 Newton's 2nd Law 10.8 Work and Power 11.1 Rolling Motion 11.2-3 Angular Momentum 11.4 Gyroscopes
Rotational Dynamics - Preview
Rotational Dynamics - Preview
Rotational Dynamics - Preview
Rigid Bodies We move from particle dynamics to the dynamics of extended rigid bodies. Three types of motion:
Rigid Bodies An unconstrained object (not on an axle) with no net force rotates about its center of mass. The CM remains motionless while every other point in the object undergoes circular motion around it.
Rigid Body Dynamics A wrench floats weightlessly in space. It is subjected to two forces of equal and opposite magnitude: Will the wrench accelerate? describes the dynamics of the object s center of mass (CM). The wrench won t translate, but it will begin to rotate.
Rotational Motion Review All points have the same ω and the same α.
Rotational Motion Review For constant α:
Rotational Motion Review Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B.
Rotational Motion Review Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B.
Rotational Motion Review The fan blade is speeding up. What are the signs of ω and α? A. ω is positive and α is positive. B. ω is positive and α is negative. C. ω is negative and α is positive. D. ω is negative and α is negative.
Rotational Motion Review The fan blade is speeding up. What are the signs of ω and α? A. ω is positive and α is positive. B. ω is positive and α is negative. C. ω is negative and α is positive. D. ω is negative and α is negative.
Rotational Motion Review At what time past 6:00 do the minute and hour hands overlap? A B C D E 6:30 6:31 6:32 6:33 7:00
Rotational Motion Review example: At what time past 6:00 do the minute and hour hands overlap? use with convenient units:
Rotational Motion Review example: At what time past 6:00 do the minute and hour hands overlap? answer: 32.727 min after six or at 6:32 and 43.6s
Rotational Energy The kinetic energy due to rotation is called rotational kinetic energy. Adding up the individual kinetic energies, and using vi = riω:
Rotational Energy Define the moment of inertia: Then rotational kinetic energy is The units of moment of inertia are kg m2. Moment of inertia depends on the axis of rotation. Mass farther from the rotation axis contributes more to the moment of inertia than mass nearer the axis. This is not a new form of energy, merely the familiar kinetic energy of motion written in a new way.
Moment of Inertia https://www.youtube.com/watch?v=m9wejfow5j0
Moment of Inertia I is resistance to change in rotational velocity that is, resistance to angular acceleration also called rotational inertia like mass but for rotation depends on mass and shape SI units: kg m2 for a point mass:
Moment of Inertia Easier to spin up Harder to spin up
Moment of Inertia For continuous objects, divide a solid object into many small cells of mass Δm and let Δ m 0. The moment of inertia sum becomes where r is the distance from the rotation axis. The procedure is much like calculating the center of mass. One rarely needs to do this integral see tables of I.
Moment of Inertia
Moment of Inertia Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless. A. Dumbbell A. B. Dumbbell B. C. Their moments of inertia are the same.
QuickCheck 12.4 Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless. A. Dumbbell A. B. Dumbbell B. C. Their moments of inertia are the same. Distance from the axis is more important than mass.
Parallel-Axis Theorem You do sometimes need to know the moment of inertia about an axis in an unusual position. You can find it if you know the moment of inertia about a parallel axis through the center of mass.
Rotational Energy: example
Rotational Energy: example
Rotational Energy: example
Rotational Dynamics The four forces shown have the same strength. Which force would be most effective in opening the door? A. B. C. D. Force F1 Force F2 Force F3 Force F4 E. Either F1 or F3
Rotational Dynamics The four forces shown have the same strength. Which force would be most effective in opening the door? A. B. C. D. Force F1 Force F2 Force F3 Force F4 E. Either F1 or F3 F1 exerts the largest torque about the hinge.
Torque Torque measures the effectiveness of the force at causing an object to rotate about a pivot. Torque is the rotational equivalent of force. On a bicycle, your foot exerts a torque that rotates the crank.
Torque The effectiveness of a force at causing a rotation is called torque. Torque is the rotational equivalent of force. We say that a torque is exerted about the pivot point.
Torque Mathematically, we define torque τ (Greek tau) as SI units of torque are N m. English units are foot-pounds. The ability of a force to cause a rotation depends on 1. the magnitude F of the force. 2. the distance r from the point of application to the pivot. 3. the angle at which the force is applied.
Torque Torque has a sign.
Moment Arm Torque is often calculated using the moment arm or lever arm. The torque is τ = df. But this is only the absolute value. You have to provide the sign of τ, based on which direction the object would rotate.
Torque Which third force on the wheel, applied at point P, will make the net torque zero?
Torque Which third force on the wheel, applied at point P, will make the net torque zero?
Torque - example
Torque - example
Torque - example
Torque
Gravitational Torque What is the net torque on this bar (about the balance point) due to its weight?
Gravitational Torque Treat it like 15 individual masses, each 1cm long:
Gravitational Torque The torques from the blue and green parts cancel each other.
Gravitational Torque Calculate the unbalance torque:
Gravitational Torque The result is equivalent to treating it as having all its mass concentrated at its Center of Mass:
Gravitational Torque Torque due to gravity is found by treating the object as if all its mass is concentrated at the center of mass.
nd Newton s 2 Law
nd Newton s 2 Law Newton s First Law: Newton s Third Law:
Rigid Body Dynamics describes the dynamics of an object s center of mass (CM).
Rigid Body Dynamics describes the dynamics of an object s center of mass (CM). describes the rotational dynamics of the object about its CM.
Rigid Body Dynamics In free-fall, gravity exerts zero torque.
A student gives a quick push to a puck that can rotate in a horizontal circle on a frictionless table. After the push has ended, the puck s angular speed A. Steadily increases. B. Increases for awhile, then holds steady. C. Holds steady. D. Decreases for awhile, then holds steady. E. Steadily decreases.
A student gives a quick push to a puck that can rotate in a horizontal circle on a frictionless table. After the push has ended, the puck s angular speed A. Steadily increases. B. Increases for awhile, then holds steady. C. Holds steady. D. Decreases for awhile, then holds steady. E. Steadily decreases. A torque changes the angular velocity. With no torque, the angular velocity stays the same. This is Newton s first law for rotation.
Rotational Dynamics - example