Chapter 10 Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts: Angular displacement (symbol: θ ) Average and instantaneous angular velocity (symbol: ω ) Average and instantaneous angular acceleration (symbol: α )
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Chapter 10Rotation
In this chapter we will study the rotational motion of rigid bodies about a fixed axis.
To describe this type of motion we will introduce the following new concepts: Angular displacement (symbol: θ ) Average and instantaneous angular velocity (symbol: ω ) Average and instantaneous angular acceleration (symbol: α )
The Rotational Variables
In this chapter we will study the rotational motion of rigid bodies about fixed axes.
Rigid body = an object that can rotate with all its parts locked together and without any change of its shape.
Fixed axis = an axis that does not move.
We take the zaxis to be the fixed axis of rotation. We define a reference line that is fixed in the rigid body and is perpendicular to the rotational axis. A top view is shown in the lower picture. The angular position of the reference line at any time t is defined by the angle θ(t) that the reference lines makes with the position at t = 0. The angle θ(t) also defines the position of all the points on the rigid body because all the points are locked as they rotate. The angle θ is related to the arc length s traveled by a point at a distance r from the axis via the equation
Note: The angle θ is measured in radians.
We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter. Consider the rigid body of the figure.
(103)
Algebraic sign of angular velocity
ω1
ω2
α = d2θ/dt2
or the second derivative with respect to t:
rO
A disk of radius 0.5 m rotates and the angular position is given by the following equation:
θ (t) = 1 + 0.6 t + 0.25 t2
a) Determine the angular position at the following times.
i) 2 s
ii) 5 s
b) At what time does the disk momentarily stop?
c) What is that value of θ when the disk momentarily stops?
d) What is the angular acceleration of the disk at 2 s?
e) What is the tangential acceleration of the disk at 2 s?
f) What is the radial acceleration of the disk at 2 s?
g) What is the magnitude of the net acceleration of the disk at 2 s?
a) θ (t) = 1 + 0.6 t + 0.25 t2Determine the angular position at the following times.
i) 2 s
= 1+.6(2) + .25 (2)2 = 1.2
ii) 5 s
= 1+.6(5) + .25 (5)2 = 2.25
b) At what time does the disk momentarily stop?
ω = 0, .6 + .5 t = 0
t = .6/.5 = 1.2 s
c) What is that value of θ when the disk momentarily stops?
θ (t) = 1 + 0.6 t + 0.25 t2 at t = 1.2 s
θ (t) = 1 + 0.6 (1.2)+ 0.25(1.2)2= 1.36 radians
d) What is the angular acceleration of the disk at 2 s?
α = dω/dt = .5 rad/s2 (constant)
e) What is the tangential acceleration of the disk at 2 s?
at = αr = (.5)(.5) = .25 m/s2
f) What is the radial acceleration of the disk at 2 s?
ar = ω(2)2r = (.4)2(.5) = .08 m/s2
g) What is the magnitude of the net acceleration of the disk at 2 s?
a = √( at2 + ar2) = .2625 m/s2
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