Question:

The moment of inertia of a solid flywheel about its axis is $0.1 \,kg\, m^2$. A tangential force of 2 kg wt. is applied round the circumference of the flywheel with the help of a string and mass arrangement as shown in the figure. If the radius of the wheel is 0.1 m, find the acceleration of the mass

Updated On: Jul 7, 2022
  • $163.3\, rad\, s^{-2}$
  • $16.3\, rad\, s^{-2}$
  • $81.66\, rad\, s^{-2}$
  • $8.16\, rad\, s^{-2}$
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The Correct Option is B

Solution and Explanation

Suppose a be the linear acceleration of the mass and T the tension in the string. Hence, $Mg - T = Ma \,...(i)$ Let $?$ be the angular acceleration of the flywheel. The couple applied to the flywheel is $I\alpha=TR$ or $T=\frac{I\alpha}{R}\,...\left(ii\right)$ Now we know that $a = R? \,...\left(iii\right)$ Putting $\left(ii\right)$ and $\left(iii\right)$ in e $\left(i\right),$ we get $Mg=\frac{I\alpha}{R}=MR\alpha$ $\therefore \alpha=\frac{MgR}{I+MR^{2}}\,...\left(iv\right)$ $=\frac{2\times9.8\times0.1}{0.1+2\times\left(0.1\right)^{2}}=16.33\,rad\,s^{-2}$
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Concepts Used:

Moment of Inertia

Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.

Moment of inertia mainly depends on the following three factors:

  1. The density of the material
  2. Shape and size of the body
  3. Axis of rotation

Formula:

In general form, the moment of inertia can be expressed as, 

I = m × r²

Where, 

I = Moment of inertia. 

m = sum of the product of the mass. 

r = distance from the axis of the rotation. 

M¹ L² T° is the dimensional formula of the moment of inertia. 

The equation for moment of inertia is given by,

I = I = ∑mi ri²

Methods to calculate Moment of Inertia:

To calculate the moment of inertia, we use two important theorems-

  • Perpendicular axis theorem
  • Parallel axis theorem