Dynamic assignment

1. Angular acceleration as a function of time and/or theta.

a. Initially a wheel is spinning counterclockwise at 36 rad/s. Friction is slowing the spin with a constant (clockwise) angular acceleration of 3 rad/𝑠". How many revolutions will the wheel make before it stops?

b. A wheel is initially spinning clockwise at 20 rad/s. The counterclockwise angular acceleration is given as a function of time on the graph below. What will be the angular speed and direction of the wheel after 10 seconds?

!" = 36 '()/+ , = 3 rad/+- (constant)

!" = 20 '()/+,(.)

, [rad/+0]

time [sec] 100

4

8

!""#$%&'%( )*7

2. Analytic Geometry a. Gear A is rotating counterclockwise at 5 rad/s. What is the angular speed and

direction of the rotation of Gear 2?

b. Block A is moving downward at 25 cm/s. What is the speed and direction of Block B?

!" = 5 rad/s !& = ?

(" = 8 *+

(& = 4 *+ rotating about fixed axes at their centers no slipping

A

B

!"# = 15'( !") = 6'(

!+# = 10'( !+) = 5'(

-. = 25 '(/2 -3 = ?

cables are not slipping

c. Load D is constrained to move vertically. The end of the cable is being pulled at 3.6 m/s as shown. What is the speed and direction of the load? What is the angular speed and direction of rotation of each of the three pulleys: 𝜔$, 𝜔&, 𝑎𝑛𝑑 𝜔+?

A

B

C

Load D

!" = 3.6 m/s(

)* = 0.25 m ). = 0.75 m )0 = 0.40 m

cables do not slip on pulleys

3. Rolling without Slipping

a. A ball with a radius of 0.15m is rolling down an incline without slipping. The center of the ball is accelerating down the slope at 3 m/𝑠". What is the direction and magnitude of the angular acceleration, 𝛼, of the ball?

b. A yoyo with an outer radius of 3 cm is moving downward on a string. The string unwinds from the yoyo from a spool with a radius of 0.5 cm. The top end of the string is held stationary. The center of the yoyo is accelerating downward at 60 cm/𝑠". What is the magnitude and direction of the angular acceleration, 𝛼, of the yoyo?

!

r =0.15 m " = ? rolling without slipping

!" = 3 &'

!( = 0.5 &'

,

-. = 60 cm/01

2 = ?

c. A ball with a radius of 0.4m is rolling down the side of a stationary cylinder with a radius of 1.0m, without slipping. At the time shown, the ball has a counterclockwise angular speed of 6 rad/s and a counterclockwise angular acceleration of 9 rad/𝑠". What are the normal, 𝑎-, and tangential, 𝑎., components of acceleration of the center of the ball, Point G, at the time shown?

G

! = 6 rad/s

$ = 9 rad/'(

) = 0.4 -

!

d. The small wheels, with radius of 0.1m, rotate without slipping on the inside of a

ring with a radius of 0.5m. The ring is rotating counterclockwise at 10 rad/s. The arms (connecting the centers of the small wheels) are rotating clockwise at 5 rad/s. What is the angular speed and direction of rotation of the small wheels, 𝜔&?

!"#$% = 10 rad/s

!* = ?

Wheels B roll without slipping on inside of ring

B

,"#$% = 0.5 m ,* = 0.1 m

!/"0 = 5 rad/s