Lecture 17 Chapter 10 Physics I 04.0.014 otational Motion Torque Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov013/physics1spring.html
Outline Chapter 10 otational Motion otational kinematics otational dynamics Torque
otational Motion In addition to translation, objects can rotate There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies Need to develop a vocabulary for describing rotational motion
In order to describe rotation, we need to define How to measure angles?
y Angular Position in polar coordinates Consider a pure rotational motion: an object moves around a fixed axis. arclength x Instead of using x and y cartesian coordinates, we will define object s position with:, θ in radians! Its definition as a ratio of two length makes it a pure number without dimensions. So, the radian is dimensionless and there is no need to mention it in calculations (Thus the unit of angle (radians) is really just a name to remind us that we are dealing with an angle). (Inverse) If angle is given in radians, we can get an arclength spanning angle θ
y Examples: angles in radians x arc length 4 / Apply radians! / x 360 rad rad 1 rad 360 / 57.3
Use adians to get an arclength 60 3
Now we need to introduce angular displacement, angular velocity and angular acceleration for rotational kinematic equations
Angular displacement and velocity Angular displacement: 1 The average angular velocity is defined as the total angular displacement divided by time: t The instantaneous angular velocity: d lim t0 t dt For both points, θ and t are the same so is the same for all points of a rotating object.
Vector of Angular Velocity Angular velocity, ω can be treated as a vector: ight Hand ule (convention) Curl fingers on right hand to trace rotation of an object Direction of thumb is vector direction for angular velocity. ( kˆ) +z +z ( kˆ)
ConcepTest 1 Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every seconds. Klyde s angular velocity is: t 1rev sec rad rad sec Bonnie and Klyde A) same as Bonnie s B) twice Bonnie s C) half of Bonnie s D) one-quarter of Bonnie s E) four times Bonnie s sec is the same for both rabbits The angular velocity of any point on a solid object rotating about a fixed axis is the same. Both Bonnie and Klyde go around one revolution ( radians) every seconds. Klyde Bonnie
Angular Acceleration The angular acceleration is the rate at which the angular velocity changes with time: Average angular acceleration: t t 1 1 1 t t t 1 Instantaneous angular acceleration: lim t 0 t d dt Since is the same for all points of a rotating object, angular acceleration also will be the same for all points. Thus, and α are properties of a rotating object
Vector of Angular acceleration Just as was the case for linear motion: 0 0 i f +z f i +z f i t t f i The object will be "speeding up" if the angular acceleration is in the same direction as the angular velocity. 0 Along z axis f i t t f i the object will be "slowing down" if the angular acceleration is in the opposite direction of the angular velocity. 0 Along -z axis
Now we need to introduce two useful expressions relating linear velocity and angular velocity and linear acceleration and angular acceleration
elation between linear and angular velocities v tan dl d Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration! The corresponding linear (or tangential) variables depend on the radius and the linear velocity is greater for points farther from the axis. By definition, linear velocity: In the 1 st slide, we defined: So we can write: v tan d dt v tan v tan d dt d d emember it!!!! You will use it often!!! elation between linear and angular velocities ( in rad/sec)
Example: Linear/angular velocity ope wound around a circular cylinder unwraps without stretching or slipping, its speed and acceleration at any instant are equal to the speed and tangential acceleration of the point at which it is tangent to the cylinder v tan v v tan v so v
ConcepTest Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every seconds. Who has the larger linear (tangential) velocity? Bonnie and Klyde II A) Klyde B) Bonnie C) both the same D) linear velocity is zero for both of them We already know that all points of a rotating body have the same angular velocity. But their linear speeds v will be v different because and tan Bonnie is located farther out (larger radius ) than Klyde. 1 V Klyde V Bonnie Klyde Bonnie
elation between linear and angular acceleration By definition of linear acceleration: a tan a dv dt d dt tan a tan tan Let s use: a tan v tan a Centripetal acceleration can be rewritten in term of angular velocity, vtan a v tan
Total acceleration Finally, any object that is undergoing circular motion experiences two accelerations: centripetal and tangential. Let s get a total acceleration: a a a a total tan a tan a total a tan a a total
otational kinematic equations The equations of motion for translational and rotational motion (for constant acceleration) are identical Translational kinematic equations otational kinematic equations v x v v x o o v o at 1 vot at a( x x o ) v x a o t o o t 1 t o o
otational Dynamics What causes rotation?
Torque When we apply the force, the door turns on its hinges (a turning effect is produced). In the 1st case, we are able to open the door with ease. In the nd case, we have to apply much more force to cause the same turning effect. Why? What causes rotation? Torque is a turning force (the rotational equivalent of force). It depends on force, lever arm, angle: rf sin A longer lever arm is very helpful in rotating objects. F r Torque due to a force F applied at a distance r from the pivot, at an angle θ to the radial line.
1 There are two ways of calculating Torque rf sin Let s arrange it like this: r( F sin ) F(rsin) Axis of rotation r F sin F Perpendicular component of force acting at a distance r from the axis Or Let s arrange it like this: r( F ) F( r ) Top view of a door Axis r sin r Force times arm lever extending from the axis to the line of force and perpendicular to the line of force r F F
Newton s nd law of rotation Force causes linear acceleration: (N. nd law) Torque causes angular acceleration: (otational N. nd law) F ma I Torque (rotational equivalent of force) Angular acceleration I is the Moment of Inertia (rotational equivalent of mass)
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ConcepTest 3 Using a Wrench You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in loosening the nut? A B Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case #) will provide the largest torque. C D E) all are equally effective Follow-up: What is the difference between arrangement A and D?
ConcepTest 3 The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest? Closing a Door A B C D E The torque is t = Fd sinq, and so the force that is at 90 to the lever arm is the one that will have the largest torque. (Clearly, to close the door, you want to push perpendicularly!!) So A or B? B has larger lever arm E A B C D