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Question: The potential energy of spring is given by \(U=\dfrac{1}{2}k{{x}^{2}}\), where \(x\) is extension sp...

The potential energy of spring is given by U=12kx2U=\dfrac{1}{2}k{{x}^{2}}, where xx is extension spring. Find the force associated with this potential energy in y-direction.

Explanation

Solution

The relation between force and potential energy is F=−dUdrF=-\dfrac{dU}{dr}.When we find force along x-direction then we differentiate the potential energy with respect to xx and when we find the force along yy-direction then we differentiate the potential energy with respect to yy.

Complete step by step answer:
Spring force is a type of conservative force
F=−dUdrF=-\dfrac{dU}{dr}
where, UUis potential energy and rr is position vector in the direction of conservative force FF is conservative force.

Work done by the conservative force is equal to negative of change in potential energy
w=−ΔUw=-\Delta U
Given, U=12kx2U=\dfrac{1}{2}k{{x}^{2}}
Put in the formula of force. When we calculate the force associated with this potential energy in x-direction then we take the position vector as xx. It means that we differentiate the potential energy with respect to xx.
F=−dUdxF=-\dfrac{dU}{dx}

After putting the value of UUin above equation then we get
F=−ddx(12kx2)F=-\dfrac{d}{dx}(\dfrac{1}{2}k{{x}^{2}})
⇒F=−12kddx(x2)\Rightarrow F=-\dfrac{1}{2}k\dfrac{d}{dx}({{x}^{2}})
The simple differentiation formula in the above equation is ddx(xn)=nxn−1\dfrac{d}{dx}({{x}^{n}})=n{{x}^{n-1}}.
F=−12k×2xF=-\dfrac{1}{2}k\times 2x
⇒F=−kx\Rightarrow F=-kx
Hence, the force along x-direction.

When we calculate the force associated with this potential energy in y-direction then we take the position vector as y it means that we differentiate the potential energy with respect to y.
F=−dUdyF=-\dfrac{dU}{dy}
After putting the value of UUin above equation then we get
F=−ddy(12kx2)F=-\dfrac{d}{dy}(\dfrac{1}{2}k{{x}^{2}})
⇒F=−12kddy(x2)\Rightarrow F=-\dfrac{1}{2}k\dfrac{d}{dy}({{x}^{2}})
When we differentiate x2{{x}^{2}}with respect to y then it became zero
∴F=0\therefore F=0

Hence, the force along y-direction is zero.

Note: The negative sign shows the spring force is a type of restoring force. The elastic potential energy of an undeformed spring is taken to be zero. Spring force is a conservative force and the conservative force does not dissipate the energy of the system into heat.