Question

Vibrations of a spring Suppose an object of mass \(m\) is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position \(y=0\) when the mass hangs at rest. Suppose you push the mass to a position \(y_{0}\) units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$ where \(k>0\) is a constant measuring the stiffness of the spring (the larger the value of \(k\), the stiffer the spring) and \(y\) is positive in the upward direction. Use equation (4) to answer the following questions. a. The period \(T\) is the time required by the mass to complete one oscillation. Show that \(T=2 \pi \sqrt{\frac{m}{k}}\) b. Assume \(k\) is constant and calculate \(\frac{d T}{d m}\) c. Give a physical explanation of why \(\frac{d T}{d m}\) is positive.

Short answer

Answer: The expression for the period of oscillation of the given mass-spring system is \(T=2 \pi \sqrt{\frac{m}{k}}\). The derivative of T with respect to mass is positive (\(\frac{d T}{d m} = \frac{\pi}{k\sqrt{m}}\)) because a heavier object requires more force to accelerate it, taking longer to complete a full cycle of oscillation.

Step-by-step solution

Determine the Period from the Given Function

The given function is: $$y(t) = y_0 \cos(t\sqrt{\frac{k}{m}})$$ The period of oscillation is the time it takes for the mass to complete one full cycle. For the cosine function, the period can be found by looking at the argument inside the cosine function: $$t\sqrt{\frac{k}{m}}$$ The period of the cosine function is \(2\pi\). To find the period T of the oscillation, we want this argument to equal \(2\pi\): $$t\sqrt{\frac{k}{m}} = 2\pi$$ Now, solve for T.

Solve for T

To solve for T, we can divide both sides of the equation by \(\sqrt{\frac{k}{m}}\): $$T = \frac{2\pi}{\sqrt{\frac{k}{m}}}$$ Next, multiply both the numerator and denominator by \(\sqrt{m}\): $$T=2 \pi \sqrt{\frac{m}{k}}$$ This is the expression for the period of oscillation of the given mass-spring system.

Find the Derivative of T with respect to m

Now, we need to find \(\frac{d T}{d m}\). The expression for T is: $$T=2 \pi \sqrt{\frac{m}{k}}$$ To find the derivative with respect to m, we must apply the chain rule: $$\frac{d T}{d m} = 2\pi \frac{1}{2}(\frac{m}{k})^{-\frac{1}{2}} \frac{d\frac{m}{k}}{d m}$$ The derivative of \(\frac{m}{k}\) with respect to m is \(\frac{1}{k}\): $$\frac{d T}{d m} = \pi (\frac{m}{k})^{-\frac{1}{2}} \frac{1}{k}$$ Simplify this expression: $$\frac{d T}{d m} = \frac{\pi}{k\sqrt{m}}$$

Explain why \(\frac{d T}{d m}\) is Positive

We have found the derivative of T with respect to m: $$\frac{d T}{d m} = \frac{\pi}{k\sqrt{m}}$$ Both \(\pi\) and \(k\) are positive constants, and \(m\) is the mass of the object, which is also positive. As a result, the entire expression is positive. Physically, this means that when the mass of the object increases, the period of oscillation also increases. This is because a heavier object requires more force to accelerate it; thus, it will take longer to complete a full cycle of oscillation. Hence, the derivative \(\frac{d T}{d m}\) is positive.

Key concepts

Period of Oscillation

Understanding the period of oscillation in a mass-spring system helps us grasp how springs behave over time. In our system, the period, represented as \( T \), refers to the time it takes for the mass to return to its initial position after completing one full cycle of movement. We can derive the period using the equation of motion given by:
\[ y(t) = y_0 \cos \left(t \sqrt{\frac{k}{m}} \right) \]
Here, \( t \sqrt{\frac{k}{m}} \) is the angular frequency. To find the period \( T \), this argument must equal \( 2\pi \) (a complete cycle for the cosine function). By solving,
\[ t \sqrt{\frac{k}{m}} = 2\pi \]
we derive the period formula as:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
The formula for the period tells us that longer periods occur with larger masses (\( m \)) or more flexible springs (lower \( k \)), thus showing the inherent relationship between mass and stiffness in oscillatory systems.

Spring Constant

The spring constant, denoted as \( k \), is a fundamental property of a spring that quantifies its stiffness. It appears directly in the formula determining the period of oscillation:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
A larger spring constant indicates a stiffer spring that requires more force to displace it the same amount compared to a spring with a smaller \( k \).

• When you pull or compress the spring, it tries to return to its original shape.
• A higher value of \( k \) leads to a more rapid oscillation.
• Conversely, a lower \( k \) means the spring is more flexible and extends or compresses more easily.

Stiffness influences how quickly the mass oscillates in a mass-spring system, directly affecting the frequency and speed of the motion. The spring constant is vital in understanding mechanical vibrations and structures in engineering, where its precise value ensures the desired dynamics of the system.

Mass-Spring System

A mass-spring system is a classic model in physics that demonstrates how objects move when under the influence of a spring force. This system consists of:

• A mass (\( m \)) attached to a spring, which can move up and down.
• The spring, characterized by its stiffness using the spring constant (\( k \)).
• The equilibrium position where the spring is neither compressed nor stretched.

For the mass-spring system, the governing equation is:
\[ y(t) = y_0 \cos \left(t \sqrt{\frac{k}{m}} \right) \]
Here, the dynamics are driven by the balance between the spring force (restoring force) and the mass's inertia. When displaced and released, the mass undergoes harmonic oscillation, moving through the equilibrium point back and forth. This replicated motion is vital in many applications:

• Designing suspension systems for vehicles
• Creating precise timekeeping clocks
• Understanding seismic responses in building structures

The study of mass-spring systems helps us better design and predict the behavior of various mechanical systems and structures foundational to physics and engineering.