Question

The mass of a mass-spring system is displaced and released from rest. If the 20 gm mass is observed to return to the release point every 2 seconds, determine the stiffness of the spring.

Short answer

Answer: The stiffness of the spring in the mass-spring system is approximately 9.87 N/m.

Step-by-step solution

Convert given mass to kilograms

The given mass is 20 grams, but usually, when working with equations in physics, we'll need to use the mass unit in kilograms, so we need to convert this: 1 kg = 1000 grams m = 20 gm * (1 kg / 1000 gm) = 0.02 kg

Find the period T

The period is given as the time it takes for the mass to return to its initial position. In this case, T = 2 seconds.

Use the formula for the period of oscillation

The formula for the period of oscillation (T) in a mass-spring system is given by: T = 2 * pi * sqrt(m / k) Where "m" is the mass of the object, "k" is the spring constant, and "pi" is the mathematical constant approximately equal to 3.14159. In this case, we have T = 2 seconds and m = 0.02 kg. We want to solve for k.

Rearrange the formula to solve for k

We can rearrange the formula for the period of oscillation to solve for k: k = m / (T / (2 * pi))^2

Plug in the values and compute k

Now we can plug in the values we know (T = 2 seconds and m = 0.02 kg) and calculate k: k = 0.02 kg / (2 s / (2 * 3.14159))^2 k = 0.02 kg / (1 / 3.14159)^2 k ≈ 9.87 N/m So the stiffness of the spring is approximately 9.87 N/m.

Key concepts

Spring Stiffness

Spring stiffness is a measure of how resistant a spring is to being compressed or stretched. It is denoted by the spring constant, often represented by the symbol "k". This constant tells us how much force is required to compress or extend the spring by a certain length.
Springs with a higher stiffness require more force to change their length. Therefore, understanding spring stiffness is vital when analyzing mass-spring systems, as it influences how the system behaves when it is set in motion.

• The spring constant is measured in Newtons per meter (N/m), which indicates the force needed for per unit of displacement.
• For springs in a mass-spring system, knowing the spring stiffness helps predict the system's oscillatory behavior, such as how quickly it returns to an equilibrium position.

By calculating the spring constant, you can determine how effective the spring is in resisting deformation.

Oscillation Period

The oscillation period in a mass-spring system is the time it takes for the system to complete one full cycle of motion.
This is the time needed for the mass to return to its starting point after being displaced and released. In many applications, like timing or vibration analysis, knowing the oscillation period helps in predicting the system's long-term behavior.

• It is denoted by the symbol "T" and is usually measured in seconds.
• The period is a crucial component in describing the dynamics of a mass-spring system.

The oscillation period is determined by both the mass attached to the spring and the spring's stiffness. In the given example, the mass returns to the release point every 2 seconds, which sets the period at 2 seconds.

Mass Conversion

Mass conversion is an essential step in physics problems, ensuring that all units are consistent with the International System of Units (SI).
In the given exercise, the mass was initially given in grams, which needed converting into kilograms for the calculations.

• This conversion is important because the standard unit for mass in SI is the kilogram (kg).
• 1 kilogram equals 1000 grams, thus the conversion for 20 grams was done by dividing by 1000.
• The conversion ensured that all other calculations, such as finding the spring constant, were accurate and according to standard physics equations.

It's a small but crucial detail that allows for precise calculations and understanding, especially when dealing with equations involving mass-spring systems.

Spring Constant Calculation

Calculating the spring constant is a key step to determine the spring's stiffness. In a mass-spring system, the spring constant "k" can be determined using the period of oscillation formula:
\[ T = 2 \pi \sqrt{\frac{m}{k}} \]This formula links the oscillation period "T", the mass "m", and the spring constant "k".
To solve for "k", we rearrange the formula:
\[ k = \frac{m}{\left(\frac{T}{2 \pi}\right)^2} \]In the exercise, the mass was 0.02 kg, and the period was 2 seconds. Plugging these into the rearranged formula:
1. First, calculate the expression for the period component,
\( \frac{2}{2 \pi} \approx \frac{1}{3.14159} \)
2. Next, square this result to fit in the formula.
3. The spring constant "k" becomes approximately 9.87 N/m.

This calculated spring stiffness tells us about the spring's resistance to deformation, helping us understand the spring's behavior in the system.