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Solutions to various problems related to angular velocity and acceleration. Topics include calculating angular velocity from revolutions per minute, converting radians to degrees, finding centripetal and tangential acceleration, and determining if an object will slide off a rotating surface. Each problem involves real-world scenarios such as bicycle wheels, crankshafts, and astronaut centrifuges.
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1 radian=180/π, therefore, 1.25 radians = 1. 25 × 180 / 2 π = 71. 62 ◦. b) Angular velocity = angular distance/time. The plane takes 8 hours, thus, ω = 1. 25 /(8 × 60 × 60) =
34 × 10 −^5 rad/s.
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 × 2 π = 37.7 rad/s. Tangential speed, v = rω, thus, v = (0. 56 /2) × 37 .7 = 10.56 m/s. radial acceleration, ac = v^2 r
= 397.942 m/s^2
ωf = ωi + αt
0 = 251.327 + α × 1. 5 α = − 167 .552 rad/s^2 Now, tangential acceleration, at = αr = − 167. 552 × 0 .015 = − 2 .513 m/s^2. b) The angular distance it covers before coming to a stop is given by,
θ = ωi + ωf 2
t
θ =
1 .5 = 188.496 rad
or θ = 188. 496 /(2π) = 30 rev
4 .833 rad/s. We can calculate his angular acceleration using, ωf = ωi + αt, or 4.833 = 0 + α × 34, or α = 0.142 rad/s^2. The tangential acceleration, at = αr = 0. 142 × 6 .8 = 0.9667 m/s^2. b) When the device is rotating at its top speed, the centripetal acceleration of the astronaut is given by ar = ω^2 r = 4. 8332 × 6 .8 = 158.848 m/s^2. In terms of g, it is, 158. 848 / 9 .8 = 16. 21 g.
ac = F/m = 16. 247 / 0 .485 = 33.5 m/s^2
Now, centripetal acceleration, ac =
v^2 r 33 .5 = v^22. 1 v = 8.387 m/s.
= g
v^2 = 38 × 9 .8 = 372. 4 v = 19.298 m/s.
ac =
Tmax m
= ω^2 r
50
ω^2 = 92. 593