Question

if F,m,r and I are known, find a and f in terms of the known quantites​

Answer 1

SOLUTION

TO DETERMINE

If F,m,r and I are known,find a and f in terms of the known quantities(simultaneous equation)

F - f = ma

( F + f )r = Iα

a = αr

EVALUATION

Here the given simultaneous equations are

F - f = ma - - - - - - (1)

( F + f )r = Iα - - - - - (2)

a = αr - - - - - - - (3)

From Equation 2 we get

$\displaystyle\sf{F + f= \frac{I \alpha }{r} }$

Using Equation 3 we get

$\displaystyle\sf{F + f= \frac{I a }{ {r}^{2} } \: \: \: - - - - (4)}$

Adding Equation 1 and Equation 4 we get

$\displaystyle\sf{2F =ma + \frac{I a }{ {r}^{2} }}$

$\displaystyle\sf{ \implies \: 2F =a \bigg(m + \frac{I }{ {r}^{2} } \bigg)}$

$\displaystyle\sf{ \implies \: F = \frac{a}{2} \bigg(m + \frac{I }{ {r}^{2} } \bigg)}$

$\displaystyle\sf{ \implies \: a = \frac{2F}{I + m {r}^{2} } }$

$\boxed{ \: \: \displaystyle\sf{ \: a = \frac{2F}{I + m {r}^{2} } } \: \: }$

Again from Equation 1 we get

f = F - ma

$\displaystyle\sf{ \implies \: f = \frac{a}{2} \bigg(m + \frac{I }{ {r}^{2} } \bigg) - ma}$

$\displaystyle\sf{ \implies \: f = \frac{a}{2} \bigg( \frac{I }{ {r}^{2} } - m\bigg)}$

$\boxed{ \: \: \displaystyle\sf{ f = \frac{a}{2} \bigg( \frac{I }{ {r}^{2} } - m\bigg)} \: \: }$

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