Question

the formula for the moment of inertia, ( i ), for the rotation of an object is ( i = \frac{m(l^2 + w^2)}{12} ) where ( m ) is the mass of the object, ( l ) is the length of the object, ( w ) is the width of the object. what formula can be used to determine ( m ), given ( i ), ( l ), and ( w )?
a. ( m = \frac{12i}{l^2 + w^2} )
b. ( m = \frac{l^2 + w^2}{12i} )
c. ( m = \frac{12(l^2 + w^2)}{i} )
d. ( m = \frac{12i}{l^2 + w^2} ) (note: likely a typo in original, assuming d is similar to a for clarity)

Explanation:

Step1: Start with the given formula

The formula for the moment of inertia \( I \) is \( I = \frac{m(L^2 + W^2)}{12} \). We need to solve for \( m \).

Step2: Multiply both sides by 12

Multiply each side of the equation by 12 to get rid of the denominator on the right - hand side. So, \( 12I = m(L^2 + W^2) \).

Step3: Divide both sides by \( (L^2 + W^2) \)

To isolate \( m \), divide both sides of the equation \( 12I = m(L^2 + W^2) \) by \( (L^2 + W^2) \). We obtain \( m=\frac{12I}{L^2 + W^2} \).

Answer:

\( m=\frac{12I}{L^2 + W^2} \) (assuming the last option in the image is of this form, if we match with the derived formula)