10.1 Rotational variables (Page 6/11)

University physics volume 1 Page 6 / 11

We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section.

Summary

The angular position $θ$ of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
The angular velocity of a rotating body about a fixed axis is defined as $ω (rad / s)$ , the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body $ω = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d θ}{d t}$ is the derivative with respect to time of the angular position $θ$ , found by taking the limit $Δ t \to 0$ in the average angular velocity $\bar{ω} = \frac{Δ θ}{Δ t}$ . The angular velocity relates $v_{t}$ to the tangential speed of a point on the rotating body through the relation $v_{t} = r ω$ , where r is the radius to the point and $v_{t}$ is the tangential speed at the given point.
The angular velocity $\vec{ω}$ is found using the right-hand rule. If the fingers curl in the direction of rotation about a fixed axis, the thumb points in the direction of $\vec{ω}$ (see [link] ).
If the system’s angular velocity is not constant, then the system has an angular acceleration. The average angular acceleration over a given time interval is the change in angular velocity over this time interval, $\bar{α} = \frac{Δ ω}{Δ t}$ . The instantaneous angular acceleration is the time derivative of angular velocity, $α = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d ω}{d t}$ . The angular acceleration $\vec{α}$ is found by locating the angular velocity. If a rotation rate of a rotating body is decreasing, the angular acceleration is in the opposite direction to $\vec{ω}$ . If the rotation rate is increasing, the angular acceleration is in the same direction as $\vec{ω}$ .
The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point.

Conceptual questions

A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand?

The second hand rotates clockwise, so by the right-hand rule, the angular velocity vector is into the wall.

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What is the value of the angular acceleration of the second hand of the clock on the wall?

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A baseball bat is swung. Do all points on the bat have the same angular velocity? The same tangential speed?

They have the same angular velocity. Points further out on the bat have greater tangential speeds.

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The blades of a blender on a counter are rotating clockwise as you look into it from the top. If the blender is put to a greater speed what direction is the angular acceleration of the blades?

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Problems

Calculate the angular velocity of Earth.

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A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?

$ω = \frac{2 π rad}{45.0 s} = 0.14 rad/s$

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A wheel rotates at a constant rate of $2.0 \times 10^{3} rev / min$ . (a) What is its angular velocity in radians per second? (b) Through what angle does it turn in 10 s? Express the solution in radians and degrees.

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A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?

a. $θ = \frac{s}{r} = \frac{3.0 m}{1.5 m} = 2.0 rad$ ; b. $ω = \frac{2.0 rad}{1.0 s} = 2.0 rad/s$ ; c. $\frac{v^{2}}{r} = \frac{{(3.0 m/s)}^{2}}{1.5 m} = 6.0 m / s^{2} .$

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A compact disc rotates at 500 rev/min. If the diameter of the disc is 120 mm, (a) what is the tangential speed of a point at the edge of the disc? (b) At a point halfway to the center of the disc?

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Unreasonable results. The propeller of an aircraft is spinning at 10 rev/s when the pilot shuts off the engine. The propeller reduces its angular velocity at a constant $2.0 rad / s^{2}$ for a time period of 40 s. What is the rotation rate of the propeller in 40 s? Is this a reasonable situation?

The propeller takes only $Δ t = \frac{Δ ω}{α} = \frac{0 rad/s - 10.0 (2 π) rad/s}{−2.0 rad / s^{2}} = 31.4 s$ to come to rest, when the propeller is at 0 rad/s, it would start rotating in the opposite direction. This would be impossible due to the magnitude of forces involved in getting the propeller to stop and start rotating in the opposite direction.

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A gyroscope slows from an initial rate of 32.0 rad/s at a rate of $0.700 {rad/s}^{2}$ . How long does it take to come to rest?

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On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of $ω = (25.0 t) rad / s$ for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at $t = 2.0 s$ ? (b) What is the angular acceleration?

a. $ω = 25.0 (2.0 s) = 50.0 rad/s$ ; b. $α = \frac{d ω}{d t} = 25.0 {rad/s}^{2}$

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The angular position of a rod varies as $20.0 t^{2}$ radians from time $t = 0$ . The rod has two beads on it as shown in the following figure, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at $t = 5 s ?$ (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at $t = 5 s ?$ (d) What are the tangential accelerations of the beads at $t = 5 s ?$ (e) What are the centripetal accelerations of the beads at $t = 5 s ?$

Figure is a drawing of a rod that rotates counterclockwise. Rod has two beads on it, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis.

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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