Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each ease, (a) υ = rω (b) a tan = rα (c) ω = ω 0 + αt ; (d) a tan = rω 2 ; (e) K = 1 2 I ω 2 .
Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each ease, (a) υ = rω (b) atan = rα (c) ω = ω0 + αt; (d) atan = rω2; (e) K =
1
2
I
ω
2
.
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 .
(a)
Expert Solution
To determine
The relation
v=rω is valid or not if the angular acceleration of an object is not constant.
Explanation of Solution
The relation for displacement is,
s=rθ (I)
s is displacement.
r is the radius of circular path
θ is angular distance.
Relation
v=rω is derived from the equation (I).
The relation
s=rθ doesn’t depend on whether angular acceleration is constant or not. Thus, if an object doesn’t have a constant acceleration it will not affect its velocity. Hence relation
v=rω is valid.
Conclusion:
The relation
v=rω is valid.
(b)
Expert Solution
To determine
The relation
atan=rα is valid or not if the angular acceleration of an object is not constant.
Explanation of Solution
The expression for tangential acceleration in terms of angular acceleration is,
atan=rα
atan is tangential acceleration.
α is angular acceleration.
Tangential acceleration is possessed by the object when it moves along the curve. The angular acceleration also doesn’t affect it. Thus relation
atan=rα is valid.
Conclusion:
The relation
atan=rα is valid.
(c)
Expert Solution
To determine
The relation
ω=ω0+αt is valid or not if the angular acceleration of an object is not constant.
Explanation of Solution
The expression for angular velocity is,
ω=ω0+αt.
ω0 is initial angular velocity.
t is the time.
ω is the final angular velocity.
The above expression is derived from the assumption that the angular acceleration is constant. Thus, relation
ω=ω0+αt is not valid.
Conclusion:
The relation
ω=ω0+αt is not valid.
(d)
Expert Solution
To determine
The relation
atan=rω2 is valid or not if the angular acceleration of an object is not constant.
Explanation of Solution
The expression for tangential acceleration in terms of angular velocity is,
atan=rω2
For an object that moves in a circular path then it has centripetal acceleration and it doesn’t depends on the whether angular acceleration is constant or not. Thus above relation is valid. Hence the relation
atan=rω2 is valid.
Conclusion:
The relation
atan=rω2 is valid.
(e)
Expert Solution
To determine
The relation
K=12Iω2 is valid or not if the angular acceleration of an object is not constant.
Explanation of Solution
The expression for kinetic energy is,
K=12Iω2 (II)
K is kinetic energy.
I is moment of inertia.
The equation (II) is derived from,
K=12mv2
m is mass.
Substitute
rω for
v in above expression to find
K.
K=12m(rω)2=12mr2ω2=12Iω2
The relation
rω is valid for any acceleration. Thus
K=12Iω2 is valid.
Conclusion:
The relation
K=12Iω2 is valid.
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A rod is rotating in a plane. The following table gives the angle (radian) at different intervals of t(sec). Calculate the angular velocity at the instant t=0.6
t
0
.2
.4
.6
.8
1
1.2
theta
0
.122
.493
1.12
2.022
3.2
4.666
A solid cylinder has a mass 2.45kg and a radius 0.746 m. What is its moment of inertia (in unit of kg.m2) when it is rotating around its own central axis? (please search your physics text book or other sources for the correct formula for this situation, you lab manual does not necessarily have the formula.)
A wheel rotates with a constant angular velocity of 2.00 rad/s
Part A
Compute the radial acceleration of a point 0.750 mm from the axis, using the relation arad=ω2rarad=ω2r.
Express your answer in meters per second squared.
Part B
Find the tangential speed of the point.
Express your answer in meters per second.
Part C
Compute the radial acceleration of the point from the relation arad=v2/rarad=v2/r.
Express your answer in meters per second squared.
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