To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration . In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h . Slum that the radius r of a given portion of the track is given by r = r i + h θ 2 π where r i is the radius of the innermost portion of the track and θ is the angle through which the disc turns to arrive at the location of the track of radius r . (b) Show that the rate of change of the angle θ is given by d θ d t = v r i + ( h θ / 2 π ) where v is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle θ as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration . In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h . Slum that the radius r of a given portion of the track is given by r = r i + h θ 2 π where r i is the radius of the innermost portion of the track and θ is the angle through which the disc turns to arrive at the location of the track of radius r . (b) Show that the rate of change of the angle θ is given by d θ d t = v r i + ( h θ / 2 π ) where v is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle θ as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
Solution Summary: The author explains that the radius of a given portion of the track is given by r=r_i+htheta 2pi
To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h. Slum that the radius r of a given portion of the track is given by
r
=
r
i
+
h
θ
2
π
where ri is the radius of the innermost portion of the track and θ is the angle through which the disc turns to arrive at the location of the track of radius r. (b) Show that the rate of change of the angle θ is given by
d
θ
d
t
=
v
r
i
+
(
h
θ
/
2
π
)
where v is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle θ as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.
Definition Definition Rate of change of angular velocity. Angular acceleration indicates how fast the angular velocity changes over time. It is a vector quantity and has both magnitude and direction. Magnitude is represented by the length of the vector and direction is represented by the right-hand thumb rule. An angular acceleration vector will be always perpendicular to the plane of rotation. Angular acceleration is generally denoted by the Greek letter α and its SI unit is rad/s 2 .
Consider again the pairs of angular positions for the rigid object in the given problem. If the object starts from rest at the initial angular position, moves counterclockwise with constant angular acceleration, and arrives at the final angular position with the same angular speed in all three cases, for which choice is the angular acceleration the highest?
You are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 19-m-tall building and land on the ground safely at a final vertical speed of 5 m/s. At the edge of the building's roof, there is a 100-kg drum that is wound with a sufficiently long rope (of negligible mass), has a radius of 0.4 m, and is free to rotate about its cylindrical axis with a moment of inertia I0. The script calls for the 76-kg stuntman to tie the rope around his waist and walk off the roof.
(a) Determine an expression for the stuntman's linear acceleration in terms of his mass m, the drum's radius r, and moment of inertia I0.
(b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of 5 m/s.
If a rigid body has a constant angular acceleration, what is the functional form of the angular velocity in terms of the time variable?
Chapter 10 Solutions
Physics for Scientists and Engineers, Technology Update, Hybrid Edition (with Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics)
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