Fisica

Chapter 11. Rotation of Rigid Bodies

Uniform Angular Acceleration
11-1. A cable is wrapped around a drum 80 cm in diameter. How many revolutions of this drum will cause an object attached to the cable to move a linear distance of 2 m?
What is the angular displacement? [ R = 0.04 m, s = 2 m, ( = ? ]
[pic] = 5 rad [pic]; ( = 0.796 rev

11-2. A bicycle wheelis 26 in. in diameter. If the wheel makes 60 revolutions, what linear distance will it travel? [ D = 26 in.; R = 13 in. = 1.083 ft ]
[pic]; s = (R = (377 rad)(1.083 ft); s = 408 ft

11-3. A point on the edge of a large wheel of radius 3 m moves through an angle of 370. Find the length of the arc described by the point.
[pic]; s = (R = (0.646 rad)(3 m); s = 1.94 m

11-4. Aperson sitting on the edge of a 6-ft diameter platform moves a linear distance of 2 ft. Express the angular displacement in radians, degrees, and revolutions. [R = 3 ft ]
[pic]; ( = 0.667 rad ; [pic]; ( = 0.106 rad ;
[pic]; ( = 38.20
11-5. An electric motor turns at 600 rpm. What is the angular velocity? What is the angular displacement after 6 s?
[pic]; ω = 62.8 rad/s[pic]; ( = 377 rad

11-6. A rotating pulley completes 12 rev in 4 s. Determine the average angular velocity in rev/s, rpm, and in rad/s?
[pic]; f = 3.00 rev/s ; [pic]; f = 28.6 rpm
[pic]; ω = 18.8 rad/s

11-7. A bucket is hung from a rope which is wrapped many times around a circular drum of radius 60 cm. The bucket starts from rest and is lifted to a height of20 m in 5 s. (a) How many revolutions were made by the drum.
(b) What was the average angular speed of the rotating drum?
[pic] = 33.3 rad [pic]; ( = 5.31 rev
[pic]; v = 4.00 m/s [pic]; ω = 6.67 rad/s

11-8. A wheel of radius 15.0 cm starts from rest and makes 2.00 rev in 3.00 s. (a) What is its average angular velocity in rad/s? (b) What is thefinal linear velocity of a point on the rim of the wheel? [pic][pic]; ω = 4.19 rad/s
11-8. (Cont.) The final angular speed is twice the average since ωo = 0: ωf = 8.38 rad/s
vf = ωfR = (8.38 rad/s)(0.15 m); vf = 1.26 m/s

11-9. A cylindrical piece of stock 6 in. in diameter rotates in a lathe at 800 rpm. What is the linear velocity at the surface of the cylinder? [R = D/2 = 3 in. = 0.250 ft ]
[pic]; v = 20.9 ft/s


11-10. The proper tangential velocity for machining steel stock is about 70 cm/s. At what rpm should a steel cylinder 8 cm in diameter be turned in a lathe? R = (0.08 m/2) = 0.04 m
[pic]; [pic];
[pic]; f = 0.279 rev/s(60 s/min); f = 16.7 rpm
11-11. For the wheel in Problem 11-8, what is the angular acceleration?What is the linear acceleration of a point on the edge of the wheel? [ ωf = 8.38 rad/s; ωo = 0, t = 3 s ]
[pic]; α = 2.79 rad/s2
a = αR = (2.79 rad/s2)(0.15 m); a = 0.419 m/s2
11-12. A circular drum of radius 40 cm is initially rotating at 400 rpm. It is brought to a stop after making 50 revolutions. What was the angular acceleration and the stopping time?
2αθ ’ ωf2 - ωo2; ( =50 rev(2( rad/rev) = 314 rad; f = 400 rpm = 41.9 rad/s
[pic]; α = 2.79 rad/s2
[pic]; t = 15.0 s
11-13. A belt is wrapped around the edge of a pulley that is 40 cm in diameter. The pulley rotates with a constant angular acceleration of 3.50 rad/s2. At t = 0, the rotational speed is 2 rad/s. What is the angular displacement and angular velocity of the pulley 2 s later?
θ ’ ω0t +½αt2; R = 0.40 m/2 = 0.20 m; ωo = 0, t = 2 s, α = 3.5 rad/s.
[pic]θ ’ ω0t + ½αt2 = (2 rad/s)(2 s) + ½(3.5 rad/s2)(2 s)2; ( = 11.00 rad
ωf = ωo + αt = 2 rad/s + (3.5 rad/s2)(2 s); ωf = 9.00 rad/s

11-14. In Problem 11-13, what is the final linear speed and linear acceleration of the pulley belt as it moves around the edge of the pulley? (We must use the radius R = 0.20 m.)
v =...