Ideal spring: find the work during certain displacement

[pedro_infante]

1. Homework Statement [/b]
A block with mass (m) is attached to an ideal spring that has force constant (k) .

2. Homework Equations [/b]

a.)The block moves from (x_1) to (x_2) , where (x_2) > (x_1) . How much work does the spring force do during this displacement?

b.The block moves from (x_1) to (x_2) and then from (x_2) to (x_1). How much work does the spring force do during the displacement from (x_2) to (x_1) ?

c.)What is the total work done by the spring during the entire (x_1) --> (x_2) --> (x_1) displacement?

d.)The block moves from (x_1) to (x_3) , where (x_3) > (x_1) . How much work does the spring force do during this displacement?

e.)The block then moves from (x_3) to (x_2) . How much work does the spring force do during this displacement?

f.) What is the total work done by the spring force during the (x_1) --> (x_3) -->(x_2) displacement?

I have no memory of ever seeing a problem like this in class. Can anyone help me?

 

[hellais]

Try drawing a graph of the displacement over the Force, and try confronting it with other similar graphs.
Consider that Work=Force*displacement.

 

[kerimek]

I can provide a response to this content by using the principles of work and energy. Work is defined as the product of force and displacement in the direction of the force. In this case, the force acting on the block is the spring force, given by Hooke's Law: F = -kx, where k is the force constant and x is the displacement from the equilibrium position.

For part a), the work done by the spring force during the displacement from (x_1) to (x_2) can be calculated by integrating the force with respect to displacement: W = ∫ Fdx = ∫ (-kx)dx = (-1/2)kx^2. Plugging in the values of x_2 and x_1, we get the work done by the spring force as (-1/2)k(x_2^2 - x_1^2).

For part b), we can use the same formula above but with the limits of integration reversed, since the block is moving in the opposite direction. The work done by the spring force during the displacement from (x_2) to (x_1) is (-1/2)k(x_1^2 - x_2^2).

For part c), we can simply add the work done in parts a) and b) to get the total work done by the spring force during the entire (x_1) --> (x_2) --> (x_1) displacement.

For part d), we can use the same formula as in part a) but with the limits of integration from x_1 to x_3. This gives us the work done by the spring force during the displacement from (x_1) to (x_3) as (-1/2)k(x_3^2 - x_1^2).

For part e), we can use the same formula as in part b) but with the limits of integration from x_3 to x_2. This gives us the work done by the spring force during the displacement from (x_3) to (x_2) as (-1/2)k(x_2^2 - x_3^2).

For part f), we can simply add the work done in parts d) and e) to get the total work done by the spring force during the (x_1) --> (x_3) -->(x