Ch. 8 Rotational Kinematics PLVs
(a) _______ when ccw. (b) _______ when cw.
(a) positive (b) negative
[8.1] A meter stick rotates about one end through an angle of 57.3°. What is the arc length traced out by the opposite end?
1 m
360/2pi = 57.3 degrees
1 radian
[8.4] An airplane prop has a radius of 1.5 m. If it starts from rest and accelerates uniformly to an angular speed of 55 rad/s in a time of 3.0 s, what is the tangential acceleration of the tip of the prop?
= (1.5 m (55rad/s) - 0) / (3.0 s) 28 m/s^2
When a rigid body rotates about a fixed axis, the angular displacement (delta theta) is the angle swept out by a line passing through any points on the body and intersecting the axis of rotation perpendicularly. (delta theta = final theta - initial theta)
Angular Displacement
[8.2] The initial angular velocity of a ceiling fan is 6.28 rad/s. It accelerates uniformly to 3.14 rad/s in 4.0 s. What is the angular acceleration?
Subtract final angular velocity (3.14 rad/s) - initial angular velocity (6.28 rad/s) over time (4.0s) Answer: -0.79 rad/s^2
[8.2] What is the average angular velocity of the seconds hand of a clock?
The seconds hand of a clock completes one rotation in 1 minute i.e 60 seconds. Angular speed=angle swept by the radius vector÷time taken Angle swept for one complete rotation is 2π radians Therefore, 2π/60=π/302π/60=π/30 Answer: -π/30 radians per second or -0.10pi radians/s Negative because of the clockwise rotation.
[8.5] In which of the following is a tangential acceleration involved in the motion? (a) A 1-kg mass swinging freely at the end of a rope (b) The tip of a lawn mower blade running at constant 40 rev/s (c) The outer edge of merry-go-round ride rotating at a constant 30o/s (d) A runner running at a constant 4 m/s around a circular track
a the other answer choices mention constant
[8.3] Which of the following sets of variables are suitable for direct substitution into the angular kinematic equations? (a) ω0= 0 rad/s, ω= 5.0 rad/s, t= 3.0 s (b) ω0= 2.0°/s, α= 2.5 rad/s2, θ= 25° (c) ω= 5.0 rev/s, α= 2.5 rad/s2, θ= 25° (d) ω0= 0 rev/s, ω= 5.0 rad/s, t= 4.0 s
a unit for angular velocity is rad/s
[8.6] Suppose the car now accelerates from 0 m/s to 30.0 m/s in 5.00 s. If the wheels have a radius of 24.1 cm, what is their angular acceleration? (a) 125 rad/s2 (b) 25.0 rad/s2 (c) 0.25 rad/s2 (d) 6.00 rad/s2
acceleration = velocity / time 30 m/s / 5 s = 6 m/s^2 tangential acceleration = radius times angular acceleration tangential acceleration / radius 6 m/s^2 / 0.241 m = 25.0 rad/s^2 b
[8.3] Staring from rest, a lawn mower blade undergoes a constant acceleration of 125.0 rad/s2. How many revolutions does it undergo in 2.0 s?
after entering it in to rotational kinematic equation, theta = 250 radians 250 radians / (2pi) = 39.78 rev Answer: 40 rev
[8.6] A car is moving with a constant speed of 70.0 mi/h (31.2 m/s). If its wheels have a radius of 24.1 cm, what is their angular speed? (a) 1.30 rad/s (b) 130 rad/s (c) 65.0 rad/s (d) 204 rad/s
angular speed = tangential velocity / radius = 31.2m/s / .241 m b
360 degrees in a full circle on revolution in 360 degrees
angular units
[8.5] What is the magnitude of the total acceleration of the tip of a 1.20-m airplane propeller that is moving with an angular speed of 35.0 rad/s and undergoing an angular acceleration of 11.0 rad/s2? (a) 42.0 m/s2 (b) 151 m/s2 (c) 138 m/s2 (d) 145 m/s2
centripetal acceleration = (1.20m) (11.0 rad/s^2)^2 = 145.2 m/s tangential acceleration = (1.20 m) (35 rad/s) = 42 m/s square root 145.2^2 + 42^2 = 151 m/s ^2 b
[8.4] If the angular speed of a 1.1-m baseball bat is 2.0 rad/s, what is the tangential speed of the tip of the bat?
v sub t = radius times angular velocity = 1.1 (2.0) 2.2 m/s
[8.1] A lawn mower blade of length 0.50 m rotates about its center. Are the magnitudes of the linear displacement (Δx) and angular displacement (Δθ) of one of the tips upon one complete rotation?
x=0 and theta=2pi