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 $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right)$, 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.
c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ($k$ is increased by a factor of $4$)?

Verified step by step guidance

1. Step 1: Start by recalling the position function of the mass on the spring, which is given by y = y_0 \cos\left(t\sqrt{\frac{k}{m}}\right).
2. Step 2: To find the velocity, differentiate the position function with respect to time t. Use the chain rule for differentiation: \frac{dy}{dt} = -y_0 \sin\left(t\sqrt{\frac{k}{m}}\right) \cdot \frac{d}{dt}\left(t\sqrt{\frac{k}{m}}\right).
3. Step 3: Differentiate the inner function t\sqrt{\frac{k}{m}} with respect to t, which results in \sqrt{\frac{k}{m}}.
4. Step 4: Substitute the derivative of the inner function back into the velocity expression: \frac{dy}{dt} = -y_0 \sin\left(t\sqrt{\frac{k}{m}}\right) \cdot \sqrt{\frac{k}{m}}.
5. Step 5: Consider the effect of increasing the stiffness k by a factor of 4. Substitute 4k for k in the velocity expression: \frac{dy}{dt} = -y_0 \sin\left(t\sqrt{\frac{4k}{m}}\right) \cdot \sqrt{\frac{4k}{m}}. Simplify the expression to see how the velocity changes.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the context of a mass-spring system, the motion is characterized by a restoring force proportional to the displacement from equilibrium, leading to sinusoidal motion. The position of the mass can be described using trigonometric functions, such as cosine, which reflects the oscillatory nature of the system.

Spring Constant (k)

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It quantifies the force required to stretch or compress the spring by a unit distance. A higher value of 'k' indicates a stiffer spring, which results in faster oscillations and greater restoring force when the spring is displaced from its equilibrium position.

Velocity in SHM

In Simple Harmonic Motion, the velocity of the oscillating object varies with time and is derived from the position function. The velocity can be expressed as the derivative of the position with respect to time. When the spring constant 'k' increases, the angular frequency of the motion increases, leading to a higher maximum velocity during oscillation, as the object moves more quickly through its equilibrium position.