Question

These exercises deal with undamped vibrations of a spring-mass system, $$ m y^{\prime \prime}+k y=0, \quad y(0)=y_{0}, \quad y^{\prime}(0)=y_{0}^{\prime} . $$ Use a value of \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) or \(32 \mathrm{ft} / \mathrm{sec}^{2}\) for the acceleration due to gravity. A 20-kg mass was initially at rest, attached to the end of a vertically hanging spring. When given an initial downward velocity of \(2 \mathrm{~m} / \mathrm{s}\) from its equilibrium rest position, the mass was observed to attain a maximum displacement of \(0.2 \mathrm{~m}\) from its equilibrium position. What is the value of the spring constant?

Short answer

Answer: The spring constant (k) for the given spring-mass system is 980 N/m.

Step-by-step solution

Calculate the force due to gravity

The formula for gravitational force is given by: \(F_{g} = m \times g\) where g is the acceleration due to gravity (9.8 \(m/s^2\)). Using this formula, we can calculate the gravitational force acting on the mass: \(F_{g} = (20 kg) \times (9.8 m/s^2) = 196 N\)

Calculate equivalent spring force at maximum displacement

At the maximum displacement, the force due to gravity and the force due to the spring are equal: \(F_{g} = k \times y_{max}\) Using the force due to gravity calculated in step 1 and the given maximum displacement: \(196 N = k \times (0.2 m)\)

Determine the spring constant

Now, we will solve for the spring constant (k). \(k = \frac{196 N}{0.2 m} = 980 N/m\) The spring constant (k) for the given spring-mass system is 980 N/m.

Key concepts

Spring-Mass System

A spring-mass system is a simple mechanical model that helps us understand how forces and motions interact. Imagine you have a spring that is either compressed or stretched, and a mass hanging or attached to its end. The key components of this system are:

• The spring, which always tries to return to its original shape. It can either push back against compression or pull back when it is stretched.
• The mass attached, which will cause the spring to extend or compress based on its weight and any motion applied to it.
• The interaction between mass and spring, which leads to vibrations or oscillations in the system.

In an undamped system, like the one described in our exercise, there's no energy loss over time. That means the mass will continue to vibrate endlessly back and forth unless an external force, like friction or air resistance, is added. Understanding this system provides a foundation for more complex physics concepts like damping, frequency, and amplitude of oscillations.

Spring Constant Calculation

The spring constant, represented as "k," is a measure of how stiff a spring is. It tells you how much force is needed to stretch or compress the spring by a certain amount. In simpler terms, a high spring constant means the spring is very stiff, and a lower spring constant means it is more flexible.Calculating the spring constant involves measuring the force applied to the spring and the resulting displacement (change in length). In the context of our exercise, this calculation was done by:

• Identifying the force due to gravity acting on the mass, which was found to be 196 N.
• Using the maximum displacement of the spring, given as 0.2 m, to find the equivalent force required.
• Applying the formula: \[k = \frac{F}{y_{max}} = \frac{196 N}{0.2 m} = 980 N/m \]

This gives a spring constant of 980 N/m, indicating the spring's resistance to deformation under the mass attached.

Gravitational Force

Gravitational force is the force that pulls objects toward the center of the Earth. It is a fundamental force that affects every object with mass. When dealing with a spring-mass system, the gravitational force (\(F_g\)) on the mass is calculated using the formula:\[ F_g = m \times g \]where:

• \(m\) is the mass in kilograms (kg).
• \(g\) is the acceleration due to gravity, typically \(9.8 m/s^2\) on Earth.

In our exercise, a 20-kg mass is subjected to gravity, resulting in a gravitational force of:\[ F_g = 20 \text{ kg} \times 9.8 \text{ m/s}^2 = 196 \text{ N} \]This force pulls the mass downward, dictating how far the spring stretches or compresses when a mass is attached. Understanding gravitational force is essential for predicting how objects behave under the influence of gravity, whether they are falling, standing still, or hanging from a spring.