*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 thepoint 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 givenvariables 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 toreach 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
⎡L⎤ ⎣ ⎦
2 or, because ω0 = 0,
ω 2 = 2α∆θ ω 2 = 2α∆θ10
(2ω)2 = 2α∆θ2ω
624 Chapter 9
The number of additional revolutions is:4∆θ10 − ∆θ10 = 3∆θ10 = 3(10 rev ) = 30 rev
and (c) is correct.
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 theproduct of a constant that is characteristic of the object’s distribution of matter, the mass of the object, and the square of the distance from the object’s center of mass to the axis about which the object is rotating. Because both (b) and (c) are correct (d ) is correct. *6 • Determine the Concept Yes. A net torque is required to change the rotational state of an object. In the absence of a nettorque an object continues in whatever state of rotational motion it was at the instant the net torque became zero. 7 • Determine the Concept No. A net torque is required to change the rotational state of an object. A net torque may decrease the angular speed of an object. All we can say for sure is that a net torque will change the angular speed of an object. 8 • (a) False. The net torque acting onan object determines the angular acceleration of the object. At any given instant, the angular velocity may have any value including zero. (b) True. The moment of inertia of a body is always dependent on one’s choice of an axis of rotation. (c) False. The moment of inertia of an object is the product of a constant that is characteristic of the object’s distribution of matter, the mass of theobject, and the square of the distance from the object’s center of mass to the axis about which the object is
rotating. 9 • Determine the Concept The angular acceleration of a rotating object is proportional to the net torque acting on it. The net torque is the product of the tangential force and its lever arm. Express the angular acceleration of the disk as a function of the nettorque acting on it: Because α ∝ d , doubling d will double the angular acceleration.
I i.e., α ∝ d
Fd F = d I I
(b) is correct.
*10 • Determine the Concept From the parallel-axis theorem we know that I = I cm + Mh 2 , where Icm is the moment of inertia of the object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance...