- Slides: 23
Angular Position, Velocity and Acceleration AP Physics C Mrs. Coyle
Rigid Body - An object or system of particles in which distances between the particles are constant. There is no deformity in the object. - Real object have some deformity.
Translation, Rotation, Rolling • Translational motion: all particles in the object have the same instantaneous velocity (linear motion) • Rotational motion: all particles in the object have the same instantaneous angular velocity • Rolling motion: combination of translation and rotation
Polar Coordinates • Radius r • Angle q measured counterclockwise from the + x axis r
-Angular position, q is positive counterclockwise from the + x axis O is the point through which the axis of rotation passes. r
Angular Displacement Dq=q-qo (Final Angle-Initial Angle) r In this figure qo=0
Units for Angular Displacement • Radian – One full revolution is 2 p radians. – Radian is actually unitless. – The radian is used in the angular kinematics equations. • Degree – 60 minutes in 1 degree, 60 seconds in 1 minute – The degree is not used in the angular kinematics equations.
Radian • Radian (rad) is the angle subtended by an arc length, s, equal to the radius. • When s=r, q=1 rad s=r r
How many radii lengths fit in an arc length s? q= s r s=qr q in radians r meters s meters • Why is the radian actually unitless?
1 rad = q [rad] = = 57. 3° [degrees]
Average Angular Speed SI unit : or rad/sec 1/sec or sec-1 • Note: common unit rpm (revolutions per minute) • Angular speed will be positive if q is increasing (counterclockwise)
Instantaneous Angular Speed
Average Angular Acceleration
Instantaneous Angular Acceleration • Units: rad/s² or s-2 • Angular acceleration is positive if an object rotating counterclockwise is speeding up or if an object rotating clockwise is slowing down.
Right Hand Rule • w, a are the magnitudes of the velocity and acceleration vectors. • The directions are given by the righthand rule.
Right Hand Rule Wrap the four fingers in the direction of rotation. The thumb shows the direction of the angular velocity vector. + angular velocity for counterclockwise rotation. - angular velocity for clockwise rotation.
Rotational Kinematic Equations
Notes – In solving rotational motion problems you must chose a rotational axis. – The object may return to its original angular position.
Example #1 The angular position of a swinging door is described by θ= 5+10 t+2 t 2, θ is in radians and t is in sec. Detremine the angular position, angular speed angular acceleration of the door a)at t=0 and b) at t=3 s. Ans: a)5 rad, 10 rad/s, 4 rad/s 2 , b)53 rad, 22 rad/s, 4 rad/s 2
Example #6 A centrifuge in a medical lab rotates at an angular speed of 3600 rev/min. When switched off, it rotates 50 times before coming to rest. Find the constant angular acceleration of the centrifuge. Ans: α= -2. 26 x 10 2 rad/s 2
Example #9 • A) Find the angular speed of the Earth’s rotation on its axis. As the Earth turns towards the east, we see the sky turning towards the west at this same rate. • B) Cambridge, England is at longitude 0 o and Saskatoon, is at longitude 107 o west. How much time elapses after the Pleiades set in Cambridge until these stars fall below the western horizon in Saskatoon? • Ans: a)7. 27 x 10 -5 rad/s, b) 2. 57 x 10 4 s or 428 min.
Example #10 A dog is running on the ground just outside a stationary merry-go-round with a constant angular speed of 0. 750 rad/s. The dog sees a bone resting on the edge of the merry-go-round one third of a revolution in front of him. At the instant the dog sees the bone (t=0) the merry-go-round begins to move in the direction the dog is running with a constant angular acceleration of 0. 015 rad/s 2. a)At what time will the dog reach the bone? (A: 2. 88 s) b)If the dog runs past the bone how long after t=0 does the dog reach the bone a second time? (Ans: 12. 8 s)