Purpose
To analyze spring displacement and develop a
mathematical model describing the relationship
between spring force and the distance stretched.
Calculate the force constant of the spring
Apply the mathematical model to determine an
expression for the potential energy of the spring.
mathematical model describing the relationship
between spring force and the distance stretched.
Calculate the force constant of the spring
Apply the mathematical model to determine an
expression for the potential energy of the spring.
Materials
Spring
masses
Triple beam balance
Meter stick
Ring stand and mounting clamp for spring
masses
Triple beam balance
Meter stick
Ring stand and mounting clamp for spring
Procedure
First chose a spring. The spring should be able to hold 5 different masses before it is stretched to the ground or table.
Once you have chosen a spring attach it to the triple beam balance using the ring stand and mounting clamp.
After you attach the spring to the triple beam balance you will chose 5 masses to hang from the spring. Make sure the heaviest one will not cause the spring to stretch to the ground and that the masses are all heaviest to measure how far the spring stretches accurately. For our lab we chose 50g 100g 150g 200g and 250g.
Hang each mass and measure how far the spring stretches for each mass. Our displacements are shown in the picture below.
Once you measure the displacements use F=ma to find the force. Since gravity and the spring force are the only 2 forces acting on the masses the acceleration is 9.8 which gives us F=m*(9.8)
Once you have chosen a spring attach it to the triple beam balance using the ring stand and mounting clamp.
After you attach the spring to the triple beam balance you will chose 5 masses to hang from the spring. Make sure the heaviest one will not cause the spring to stretch to the ground and that the masses are all heaviest to measure how far the spring stretches accurately. For our lab we chose 50g 100g 150g 200g and 250g.
Hang each mass and measure how far the spring stretches for each mass. Our displacements are shown in the picture below.
Once you measure the displacements use F=ma to find the force. Since gravity and the spring force are the only 2 forces acting on the masses the acceleration is 9.8 which gives us F=m*(9.8)
Analysis
We know that F(x)= -dU/dx and we can rearrange this to dU/dx= -F(x). F(x)= -kx
To find the potential energy in the spring with respect to the displacement we know that F(x)= -dU/dx and F(x)= -kx so -dU/dx= -kx or dU/dx=kx. we take the anti derivative of dU/dx=kx to find that U= (1/2)*kx^2 which is the equation for an ideal spring.
The spring constant is the increase in the force F with respect to the position x or just the derivative of F(x) which is the slope. As shown in the graph above the spring constant (slope of the force vs displacement line) for our spring was 28.481.
One unusual result we found is that the y intercept is not zero. We decided the most likely cause of this was the force of the weight of the spring itself. We also found that our spring was ideal which we were told was unlikely.
To find the potential energy in the spring with respect to the displacement we know that F(x)= -dU/dx and F(x)= -kx so -dU/dx= -kx or dU/dx=kx. we take the anti derivative of dU/dx=kx to find that U= (1/2)*kx^2 which is the equation for an ideal spring.
The spring constant is the increase in the force F with respect to the position x or just the derivative of F(x) which is the slope. As shown in the graph above the spring constant (slope of the force vs displacement line) for our spring was 28.481.
One unusual result we found is that the y intercept is not zero. We decided the most likely cause of this was the force of the weight of the spring itself. We also found that our spring was ideal which we were told was unlikely.
Conclusion
Our mathematical model for the spring force and displacement was F(x)= -kx
We found the force constant k for our spring to be 28.481.
We also found the equation of the potential energy U as a function of distance x to be U = (1/2)*kx^2
We found the force constant k for our spring to be 28.481.
We also found the equation of the potential energy U as a function of distance x to be U = (1/2)*kx^2