HW 6 1
Homework 6
1. 1 revolution = 2πradians and 1 minute = 60 seconds. Therefore, 56 revolutions per minute = 56 ×2π/60
rad/s= 5.86 rad/s.
2. a)x=rθ, or θ=x/r radians. Therefore, θ= 5000/4000 = 1.25 radians.
1 radian=180/π, therefore, 1.25 radians = 1.25 ×180/2π= 71.62◦.
b) Angular velocity = angular distance/time. The plane takes 8 hours, thus, ω= 1.25/(8 ×60 ×60) =
4.34 ×10−5rad/s.
3. Your roommate is working on his bicycle and has the bike upside down. He spins the 56 cm (or 0.56 m)
diameter wheel, and a pebble in the tread goes by 6 times every second. Thus, the wheel’s angular speed
is 6 rev/s or 6 ×2π= 37.7 rad/s. Tangential speed, v=rω, thus, v= (0.56/2) ×37.7 = 10.56 m/s.
radial acceleration,
ac=v2
r=10.562
0.28 = 397.942 m/s2
4. Mass of the ball = 28 g = 0.028 kg
The track is 42 cm in diameter or 42/2 = 21 cm in radius.
Speed of the ball = 60 rpm = 60 ×2π/60 rad/s = 6.283 rad/s.
Centripetal acceleration of the ball= ac=ω2r= 6.2832×0.21 = 8.29 m/s2.
The force for this acceleration comes from the track. F=mac= 0.028 ×8.29N= 0.232N.
5. Diameter of the crankshaft = 3.0 cm. Therefore, radius = 3.0/2 = 1.5 cm = 0.015 m.
Initial rotation speed = 2400 rpm = 2400 ×2π/60 rad/s = 251.327 rad/s.
The wheel comes to a stop in 1.5 s.
a) Therefore, its angular acceleration can be calculated using,
ωf=ωi+αt
0 = 251.327 + α×1.5
α=−167.552 rad/s2
Now, tangential acceleration, at=αr =−167.552 ×0.015 = −2.513 m/s2.
b) The angular distance it covers before coming to a stop is given by,
θ=ωi+ωf
2t
θ=251.327 + 0
21.5 = 188.496 rad
or θ= 188.496/(2π) = 30 rev
6. Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge takes 34 s to
speed up from rest to its top speed of 1 rotation every 1.3 s. The radius of the circle in which the astronaut
moves is 6.8 m.
a) The astronaut’s top angular speed is 1 revolution per 1.3 s. Thus, its speed is 1 ×2π/1.3 rad/s =
