How to find spring constant from mass and displacement?
The Correct Answer and Explanation is:
To find the spring constant (( k )) from mass (( m )) and displacement (( x )), you can use Hooke’s Law and the concept of simple harmonic motion (if applicable). Here’s how to approach it:
Formula:
The spring constant can be found using Hooke’s Law, which is given by:
[
F = k \cdot x
]
Where:
- ( F ) is the force applied to the spring,
- ( k ) is the spring constant,
- ( x ) is the displacement (stretch or compression) from the spring’s equilibrium position.
In the context of a mass attached to a spring, the force exerted on the spring is typically the gravitational force acting on the mass, which is:
[
F = m \cdot g
]
Where:
- ( m ) is the mass of the object (in kilograms),
- ( g ) is the acceleration due to gravity (approximately ( 9.8 \, \text{m/s}^2 )).
Thus, by combining Hooke’s Law with the force from gravity, we can find the spring constant:
[
m \cdot g = k \cdot x
]
Solving for ( k ):
[
k = \frac{m \cdot g}{x}
]
Steps to Find the Spring Constant:
- Measure the displacement (( x )): This is how much the spring stretches or compresses from its equilibrium position when the mass is attached to it.
- Calculate the force: Multiply the mass ( m ) by the acceleration due to gravity ( g ) to find the gravitational force acting on the mass.
- Substitute values into the formula: Plug the values for ( m ), ( g ), and ( x ) into the formula ( k = \frac{m \cdot g}{x} ) to find the spring constant ( k ).
Example:
If a mass of 2 kg is attached to a spring and stretches the spring by 0.1 meters, the spring constant ( k ) would be:
[
k = \frac{2 \, \text{kg} \times 9.8 \, \text{m/s}^2}{0.1 \, \text{m}} = \frac{19.6 \, \text{N}}{0.1 \, \text{m}} = 196 \, \text{N/m}
]
Thus, the spring constant ( k ) is 196 N/m.
This method assumes the spring is in its elastic region, where Hooke’s Law is valid.