Question

A mass \(m\) is attached to a spring with a spring constant of \(k\) and set into simple harmonic motion. When the mass has half of its maximum kinetic energy, how far away from its equilibrium position is it, expressed as a fraction of its maximum displacement?

Short answer

Short Answer: When the kinetic energy of the mass is half of its maximum, its position relative to the equilibrium point is approximately 0.71 times its maximum displacement.

Step-by-step solution

Total energy in simple harmonic motion

In simple harmonic motion, the total energy (E) is the sum of potential energy (U) stored in the spring and the kinetic energy (K) of the mass. We can represent this relationship with the following equation: $$ E = U + K $$

Express the kinetic energy in terms of potential energy

The potential energy (U) stored in the spring is given by the equation: $$ U = \frac{1}{2}kx^2 $$ where \(k\) is the spring constant and \(x\) is the displacement from the equilibrium position. The kinetic energy (K) of the mass is given by the equation: $$ K = \frac{1}{2}mv^2 $$ where \(m\) is the mass and \(v\) is the velocity. We can express the kinetic energy (K) in terms of potential energy (U) as follows: $$ \frac{1}{2}mv^2 = E - \frac{1}{2}kx^2 $$

Use the given conditions to find a relationship between potential and kinetic energy

We are given that the kinetic energy of the mass is half its maximum value. In terms of potential energy, this means: $$ \frac{1}{2}mv^2 = \frac{1}{2}E = E - \frac{1}{2}kx^2 $$ From this equation, we can deduce that the potential energy is equal to half of the maximum energy in the system: $$ \frac{1}{2}kx^2 = \frac{1}{2}E $$

Determine the displacement of the mass from the equilibrium position

We are interested in finding the displacement \(x\) as a fraction of its maximum displacement \(A\). We know that at maximum displacement, the potential energy (U) is equal to the total energy (E). We can use the potential energy equation to express maximum displacement (A) as follows: $$ \frac{1}{2}kA^2 = E $$ Now, we can use the relationship between potential energy and maximum energy to write an expression for the displacement \(x\): $$ \frac{x^2}{A^2} = \frac{\frac{1}{2}kx^2}{\frac{1}{2}kA^2} = \frac{\frac{1}{2}E}{E} $$ From this equation, we can find the fraction of the maximum displacement: $$ \frac{x^2}{A^2} = \frac{1}{2} $$ Taking the square root of both sides, we get: $$ \frac{x}{A} = \frac{1}{\sqrt{2}} $$ So, when the mass has half of its maximum kinetic energy, it is at a distance of \(\frac{1}{\sqrt{2}} \(≈ 0.71\)\) times its maximum displacement from the equilibrium position.

Key concepts

Spring Constant

Understanding the spring constant is crucial when delving into simple harmonic motion. The spring constant, denoted as k, is a measure of stiffness of the spring. It's defined as the force required to stretch or compress the spring by a unit distance. In mathematical terms, if a force F causes a displacement x in the spring, then the spring constant is given by Hooke's Law which states:

\[\begin{equation}k = \frac{F}{x}\end{equation}\]
Higher values of k indicate a stiffer spring that's more resistant to deformation. This concept is intimately linked to simple harmonic motion because the force exerted by the spring is what causes the mass attached to it to oscillate. The force is also directly proportional to the displacement from the spring's equilibrium position, leading to the characteristic back-and-forth motion of the system.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. For any moving object with a mass m and velocity v, the kinetic energy K can be calculated using the following equation:

\[\begin{equation}K = \frac{1}{2}mv^2\end{equation}\]
When dealing with simple harmonic motion, kinetic energy varies throughout the cycle. It is maximum when the object passes through the equilibrium position, as all the potential energy stored in the spring at maximum displacement has been converted into kinetic energy. Conversely, the kinetic energy is zero at maximum displacement because the object momentarily comes to rest before reversing its motion.

Potential Energy

In the context of simple harmonic motion, potential energy refers to the energy stored within the spring when it is either compressed or stretched. This potential energy U is a function of the spring constant k and the displacement x from the spring's equilibrium position, represented by the equation:

\[\begin{equation}U = \frac{1}{2}kx^2\end{equation}\]
The potential energy is at its maximum when the spring is at its maximum displacement (either stretched or compressed the most). At the equilibrium position, all of this potential energy has been transformed into kinetic energy, therefore the potential energy is zero there. The interplay between kinetic and potential energy underpins the oscillatory nature of simple harmonic motion.

Equilibrium Position

The equilibrium position in simple harmonic motion is the point at which the forces acting on the mass are balanced, resulting in no net force. It is essentially the midpoint of the oscillation where the spring is neither stretched nor compressed. This position is critical because it's where the object would remain at rest if there were no external disturbances. When an object in simple harmonic motion passes through the equilibrium position, its potential energy is at its minimum (zero), and its kinetic energy is at its maximum, because all the stored elastic potential energy has been converted to kinetic energy. It's also the point where the object's velocity reaches its peak.

Maximum Displacement

Maximum displacement, often denoted as A, is the furthest distance from the equilibrium position that the mass reaches during its oscillation in simple harmonic motion. This is effectively the amplitude of the oscillation, and it corresponds to the point where the spring's potential energy is greatest and the kinetic energy is zero—because the object is momentarily stationary before reversing direction. The maximum displacement is a key factor in determining the energy of the system as well as the period and frequency of the motion. Understanding the maximum displacement is vital for solving problems related to energy and motion within the context of simple harmonic motion.