Teste
Conceptual Problems
*1 • Determine the Concept Because r is greater for the point on the rim, it moves the greater distance. Both turn through the same angle. Because r is greater for the point on the rim, it has the greater speed. Both have the same angular velocity. Both have zero tangential acceleration. Both have zero angular acceleration. Because r is greater for the point on the rim, it has the greater centripetal acceleration. 2 •
(a) False. Angular velocity has the dimensions ⎢ ⎥ whereas linear velocity has ⎣T ⎦ dimensions ⎢ ⎥ . T (b) True. The angular velocity of all points on the wheel is dθ/dt. (c) True. The angular acceleration of all points on the wheel is dω/dt. 3 •• Picture the Problem The constant-acceleration equation that relates the given variables 2 is ω 2 = ω0 + 2α∆θ . We can set up a proportion to determine the number of revolutions required to double ω and then subtract to find the number of additional revolutions to accelerate the disk to an angular speed of 2ω. Using a constant-acceleration equation, relate the initial and final angular velocities to the angular acceleration: Let ∆θ10 represent the number of revolutions required to reach an angular velocity ω: Let ∆θ2ω represent the number of revolutions required to reach an angular velocity ω: Divide equation (2) by equation (1) and solve for ∆θ2ω:
2 ω 2 = ω0 + 2α∆θ
⎡1⎤
⎡L⎤ ⎣ ⎦
2 or, because ω0 = 0,
ω 2 = 2α∆θ ω 2 = 2α∆θ10
(1)
(2ω)2 = 2α∆θ2ω
(2)
∆θ2ω =
623
(2ω)2 ∆θ ω2 10
= 4∆θ10
624 Chapter 9
The number of additional revolutions is:
4∆θ10 − ∆θ10 = 3∆θ10 = 3(10 rev ) = 30 rev and (c) is correct.
*4
•
Determine the Concept Torque has the dimension ⎢ (a) Impulse has the dimension ⎢
⎡ ML2 ⎤ . 2 ⎥ ⎣T ⎦
⎡ ML ⎤ . ⎣ T ⎥ ⎦ ⎡ ML2 ⎤ (b) Energy has the dimension ⎢ 2 ⎥ . ⎣T ⎦ ⎡ ML ⎤ . ⎣ T ⎥ ⎦
(b) is correct.
(c) Momentum has the dimension ⎢
5 • Determine the Concept The moment of inertia of an object is the