define angular displacement , angular speed and angular acceleration. How are linear velocity and linear acceleration at any insatnt
Answers
Answer:
Equations. The linear speed is proportional to the angular speed and the radius. The average angular acceleration is the change in angular velocity divided by time. The tangential acceleration is proportional to the angular acceleration and the radius.
Explanation:
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Answer:
Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s2. (b) If she now slams on the brakes, causing an angular acceleration of -87.3 rad/s2, how long does it take the wheel to stop?
Strategy for (a)
The angular acceleration can be found directly from its definition in
α
=
Δ
ω
Δ
t
because the final angular velocity and time are given. We see that Δω is 250 rpm and Δt is 5.00 s.
Solution for (a)
Entering known information into the definition of angular acceleration, we get
α
=
Δ
ω
Δ
t
=
250 rpm
5.00 s
.
Because Δω is in revolutions per minute (rpm) and we want the standard units of rad/s2 for angular acceleration, we need to convert Δω from rpm to rad/s:
Δ
ω
=
250
rev
min
⋅
2
π
rad
rev
⋅
1
min
60
sec
=
26.2
rad
s
Entering this quantity into the expression for α, we get
α
=
Δ
ω
Δ
t
=
26.2 rad/s
5.00 s
=
5.24
rad/s
2
.
Strategy for (b)
In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for Δt, yielding
Δ
t
=
Δ
ω
α
.
Solution for (b)
Here the angular velocity decreases from 26.2 rad/s (250 rpm) to zero, so that Δω is –26.2 rad/s, and α is given to be -87.3 rad/s2. Thus,
Δ
t
=
−
26.2 rad/s
−
87.3
rad/s
2
=
0.300 s.
Discussion
Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall—the velocity change is large in a short time interval.