Question

How do you solve for m in the equation $\dfrac{1}{2}m{{v}^{2}}=\dfrac{1}{2}k{{x}^{2}}$ ?

Answer 1

Hint: To solve for m in the equation $\dfrac{1}{2}m{{v}^{2}}=\dfrac{1}{2}k{{x}^{2}}$ , we have to cancel the common terms from both the sides. Then, we have to take ${{v}^{2}}$ to the RHS. We can find the value of m clearly, if $v=x$ .

Complete step-by-step solution:
We have to solve for m in the equation $\dfrac{1}{2}m{{v}^{2}}=\dfrac{1}{2}k{{x}^{2}}$ . Let us cancel $\dfrac{1}{2}$ from both the sides.
$\Rightarrow \require{cancel}\cancel{\dfrac{1}{2}}m{{v}^{2}}=\require{cancel}\cancel{\dfrac{1}{2}}k{{x}^{2}}$
We will get the result of the above simplification as
$\Rightarrow m{{v}^{2}}=k{{x}^{2}}...\left( i \right)$
Now, we have to take ${{v}^{2}}$ to the RHS. This will be the divisor in the RHS.
$\Rightarrow m=\dfrac{k{{x}^{2}}}{{{v}^{2}}}$
The value of m depends on the value of x. Let us consider equation (i).
If ${{v}^{2}}={{x}^{2}}$ or $v=x$ , we can find the value of m as
$\Rightarrow m{{v}^{2}}=k{{v}^{2}}$
Let us cancel ${{v}^{2}}$ from both the sides.
$\Rightarrow m\require{cancel}\cancel{{{v}^{2}}}=k\require{cancel}\cancel{{{v}^{2}}}$
We will get the result of the above simplification as
$\Rightarrow m=k$
Therefore, the value of m is $\dfrac{k{{x}^{2}}}{{{v}^{2}}}$ and if $v=x$ , we will get $m=k$ .

Note: Students must know to solve algebraic equations and the rules associated with it. When we move a positive term to other side, it becomes negative. Likewise, when we move a negative term to other side, it will become positive. When we move a divisor to one side, it will be multiplied with the terms on the moved side. We cannot find the value of m unless the values of k,x and v are given.The given equation is related to physics, that is, mass and energy of a spring. The kinetic energy of the spring is equal to its elastic potential energy.