Respuesta :
Answer:
50 revolutions
Explanation:
Data provided:
case I: From rest to top spin
The initial angular speed of the washer, ωi = 0 rev /s
Final angular speed of the washer ωf = 5 rev /s
Time taken, t₁ = 8 s
now,
The angular displacement or the number of revolutions taken (θ₁) is calculated as:
θ₁ = ωi t₁ + (1/2)α₁t₁²
where,
α is the angular acceleration
The angular acceleration can be calculated as:
ωf - ωi = α₁t₁
on substituting the values, we get
8α₁ = 5 - 0
or
α₁ = 0.625 rev/s²
substituting the values in the equation for the number of revolutions, we get
θ₁ = 0 + (1/2) (0.625)(8)²
or
θ₁ = 20 revolutions
also,
For the case II: From top spin to rest
we have
The initial angular speed, ωi = 5 rev /s
and the final angular speed, ωf = 0 rev /s
Total time taken, t₂ = 12 s
Now, angular acceleration for this case
ωf - ωi = α₂t₂
on substituting the values, we have
12α₂ = 0 - 5
α₂ = - 0.4166 rev/s²
Therefore, the number of revolutions ( i.e angular displacement )
θ₂ = ωit₂ + (1/2)α₂t₂²
on substituting the values, we have
θ₂ = 5 × 12 + (1/2)(-0.4166)(12)²
or
θ₂ = 30 rev
Hence,
the total number of revolutions made by the washer during the 20s is
θ = θ₁ + θ₂
or
θ = 20 rev + 30 rev
or
θ = 50 revolutions
The number of revolutions made by the tub during the entire time is 15 revolutions.
Angular acceleration of the tub
The angular acceleration of the tub is determined by applying the following kinematic equation as shown below;
[tex]\omega _f = \omega _i \ + \ a\lpha t\\\\[/tex]
where;
- ωi is the initial angular velocity
- ωf is the final angular velocity = 5 rev/s = 5 x 2π rad = 31.42 rad/s
[tex]31.42 = 0 \ + \ \alpha (8)\\\\\alpha = \frac{31.42}{8} \\\\\alpha = 3.93 \ rad/s^2[/tex]
Displacement of the tub after 20s
The displacement of the tub before it comes to rest is calculated as follows;
[tex]\theta = \omega_i t + \frac{1}{2} \alpha t^2\\\\\theta = 31.42(12) \ + \ 0.5(-3.93)(12)^2\\\\\theta = 377.04 \ - 282.96\\\\\theta = 94.1 \ rad\\\\\theta = 94.1\ rad \times \frac{1 \ rev}{2\pi \ rad} = 15 \ rev[/tex]
Thus, the number of revolutions made by the tub during the entire time is 15 revolutions.
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