A mass attached to a spring has simple harmonic motion with an amplitude of 4.0 cm. When the mass is 2.0 cm from the equilibrium position, what fraction of its total energy is potential energy? A. one-half B. one-third C. three-quarters D. two-thirds E. one-quarter
The correct answer and explanation is:
The correct answer is A. one-half.
Explanation:
In simple harmonic motion (SHM), the total mechanical energy (E) of the system remains constant and is the sum of kinetic energy (KE) and potential energy (PE). The total energy is stored in the spring when the mass is at the maximum displacement (amplitude, A), where all energy is potential, and kinetic energy is zero.
The potential energy in a spring at a displacement xx from the equilibrium position is given by: PE=12kx2PE = \frac{1}{2} k x^2
where kk is the spring constant.
The total energy is the potential energy at maximum amplitude: E=12kA2E = \frac{1}{2} k A^2
The fraction of the total energy that is potential energy at displacement xx is: PEE=12kx212kA2=x2A2\frac{PE}{E} = \frac{\frac{1}{2} k x^2}{\frac{1}{2} k A^2} = \frac{x^2}{A^2}
Given:
- Amplitude A=4.0 cmA = 4.0 \text{ cm}
- Displacement x=2.0 cmx = 2.0 \text{ cm}
Calculate the fraction: PEE=(2.0)2(4.0)2=416=14\frac{PE}{E} = \frac{(2.0)^2}{(4.0)^2} = \frac{4}{16} = \frac{1}{4}
This suggests potential energy is one-quarter of the total energy.
However, this is the fraction of total energy stored as potential energy. The problem asks, “what fraction of its total energy is potential energy when the mass is 2.0 cm from equilibrium?” The fraction is one-quarter, corresponding to option E.
Let’s carefully reconsider the options and problem wording:
- At displacement x=2.0 cmx = 2.0 \text{ cm}, the potential energy fraction is x2A2=416=14\frac{x^2}{A^2} = \frac{4}{16} = \frac{1}{4}.
- So, the potential energy is one-quarter of total energy.
- The kinetic energy is then 1−14=341 – \frac{1}{4} = \frac{3}{4}.
The correct choice is E. one-quarter.
Summary:
In simple harmonic motion, potential energy varies with the square of displacement from equilibrium relative to the square of the amplitude. At half the amplitude, potential energy is one-quarter of the total mechanical energy. This relationship helps to understand energy distribution between kinetic and potential forms during oscillation.