A record player turntable initially rotating at 3313 rev/min is braked to a stop at a constant rotational acceleration. The turntable has a rotational inertia of 0.031kg?m2 . When it is switched off, it slows down to 75% of its initial rotational speed in 4.0 s.A- How long does it take to come to rest?B- How much work has to be done on the turntable to bring it to rest?

Respuesta :

Answer:

15.99861 seconds

1865.58658 Joules

Explanation:

[tex]\alpha[/tex] = Angular acceleration

[tex]\theta[/tex] = Angle of rotation

I = Moment of inertia of disc = 0.031 kgm²

t = Time taken

Initial angular speed

[tex]\omega_i=3313\frac{2\pi}{60}\\\Rightarrow \omega_i=346.93\ rad/s[/tex]

Final angular velocity

[tex]\omega_f=0.75\times \omega_i\\\Rightarrow \omega_f=0.75\times 346.93\\\Rightarrow 260.19\ rad/s[/tex]

From the equation of rotational motion

[tex]\omega_f=\omega_i+\alpha t\\\Rightarrow \alpha=\frac{\omega_f-\omega_i}{t}\\\Rightarrow \alpha=\frac{260.19-346.93}{4}\\\Rightarrow \alpha=-21.685\ rad/s^2[/tex]

[tex]\omega_f=\omega_i+\alpha t\\\Rightarrow t=\frac{\omega_f-\omega_i}{\alpha}\\\Rightarrow t=\frac{0-346.93}{-21.685}\\\Rightarrow t=15.99861\ s[/tex]

Time taken by the turntable to slow down is 15.99861 seconds

Rotational energy is given by

[tex]K=\frac{1}{2}I\omega_i^2\\\Rightarrow K=\frac{1}{2}\times 0.031\times 346.93^2\\\Rightarrow K=1865.58658\ J[/tex]

The work has to be done on the turntable to bring it to rest is 1865.58658 Joules

Lanuel

The amount of time it would take the record player turntable to come to rest is 15.99 seconds and the amount of work done is 1865.59 Joules.

How to calculate the time?

First of all, we would determine the final angular velocity of this record player turntable as follows:

ωf = 75% × ωi

Note: ωi = 3313 × 2π/60 = 346.93 rad/s.

ωf = 0.75 × 346.93

ωf = 260.19 rad/s.

Next, we would determine the angular acceleration by using the first equation of rotational motion:

ωf = ωi + αt

α = (ωf - ωi)/t

α = (260.19 - 346.93)/4

α = -21.685 rad/s².

Similarly, the time taken is given by:

t = (ωf - ωi)/α

t = (0 - 346.93)/-21.685

Time, t = 15.99 seconds.

How to calculate the work done?

Mathematically, the rotational energy possessed by this record player turntable is equal to the work done and it can be calculated by using this formula:

K.E = 1/2 × Iωi²

K.E = 1/2 × 0.031 × 346.93²

K.E = 1865.59 Joules.

Read more on angular acceleration here: https://brainly.com/question/6860269

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