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) 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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For selected values of angular position, velocity, acceleration and a couple of times, complete the following table. Radius is 2 and all the values in S.I unit.
θo=3
ωo=4
α=5
θt
∆θt
ω
∆s
vtang
atang
t2=3
t2=5
The uniform 22-kg plate is released from rest at the
position shown. (Figure 1)
Figure
A
x
0.5 m-
1 of 1
0.5 m
>
Part A
Determine its initial angular acceleration.
Express your answer in radians per second squared to three significant figures. Enter positive value if the
angular acceleration is clockwise and negative value if the angular acceleration is counterclockwise.
a =
Submit
Part B
195| ΑΣΦ. 41
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vec
?
rad/s²
Review
Determine the x and y components of reaction at the pin A using scalar notation.
Express your answers in newtons to three significant figures separated by a comma.
Important note: Unless otherwise stated, your answer must be in SI units.
A bicycle wheel 67.0 [cm] in diameter rotates about its central axis with a constant angular acceleration of 4.48 [rad/s²]. It starts from rest at
time t = 0 [s], and a reference point on the wheel's rim P makes an angle of 60.0° above the horizontal at this time. When the wheel starts
rotating, at time t = 3.70 [s],
At t = 1.50 [s], a small, 92.5 [g] pebble got stuck on the rim of the wheel.
E What is the pebble's moment of inertia with respect to the wheel's axis of rotation?
F
At t = 3.70 [s], what is the pebble's rotational kinetic energy?
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