Angular Position, Velocity and Acceleration

AP Physics CMrs. 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 measured counterclockwise from the

+ x axis

r

-Angular position, is positive counterclockwise from the + x axis

r

O is the point through which the axis of rotation passes.

Angular Displacement

In this figure qo=0

(Final Angle-Initial Angle)

r

Units for Angular Displacement• Radian–One full revolution is 2radians.–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, =1 rad s=r

r

How many radii lengths fit in an arc length s?

= s s=rr

in radians r meters

s meters • Why is the radian actually unitless?

1 rad = = 57.3°

[rad] = [degrees]

Average Angular Speed

SI unit : rad/sec or 1/sec or sec-1

• Note: common unit rpm (revolutions per minute) • Angular speed will be positive if is increasing

(counterclockwise)

f i

f it t t

Instantaneous Angular Speed

lim0 t

d

t dt

Average Angular Acceleration

f i

f it t t

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.

lim0 t

d

t dt

Right Hand Rule• are the

magnitudes of the velocity and acceleration vectors.

• The directions are given by the right-hand 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

f i t 21

2f i it t

2 2 2 ( )f i f i

1

2f i i f t

Notes

–In solving rotational motion problems you must chose a rotational axis.

–The object may return to its original angular position.

Example #1The angular position of a swinging door is described by θ= 5+10t+2t2, θ is in radians and t is in sec. Detremine the angular position, angular speed and angular acceleration of the door a)at t=0 and b) at t=3s.

Ans: a)5 rad, 10rad/s, 4 rad/s2 , b)53 rad, 22rad/s, 4 rad/s2

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= -2.26x10 2 rad/s2

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 0o and Saskatoon, is at longitude 107o west. How much time elapses after the Pleiades set in Cambridge until these stars fall below the western horizon in Saskatoon?

• Ans: a)7.27x10 -5 rad/s, b) 2.57 x10 4 s or 428min.

Example #10A 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/s2 .

a)At what time will the dog reach the bone? (A:2.88s)b)If the dog runs past the bone how long after t=0

does the dog reach the bone a second time?(Ans:12.8s)