A 4.30 kg block starts from rest and slides down a frictionless incline, dropping a vertical distance of 2.60 m, before compressing a spring of force constant 2.20 104 N/m. Find the maximum compression of the spring.
To find the maximum compression of the spring, we can use the principle of conservation of mechanical energy. The energy at the top of the incline is solely gravitational potential energy, while at the maximum compression of the spring, the energy is purely elastic potential energy.
Here's how you can solve it step by step:
Step 1: Find the gravitational potential energy (PE) at the top of the incline:
PE = mgh
where m = mass, g = acceleration due to gravity, h = height
In this case, m = 4.30 kg, g = 9.8 m/s^2, and h = 2.60 m.
Substitute the values into the formula:
PE = (4.30 kg) x (9.8 m/s^2) x (2.60 m)
Calculate the gravitational potential energy.
Step 2: Convert the gravitational potential energy into elastic potential energy:
The total mechanical energy (E) remains constant throughout the motion.
E = PE + KE + XE
where KE is the kinetic energy and XE is the elastic potential energy.
Since the block starts from rest, KE at the top is zero.
Therefore, E = PE + XE
Step 3: Equate gravitational potential energy with the maximum elastic potential energy:
PE = XE
Substitute the calculated gravitational potential energy into the equation.
Step 4: Solve for the maximum compression of the spring (XE):
XE = (2.20 x 10^4 N/m) x (XE)^2 / 2
Rewrite the equation to solve for XE.
Step 5: Solve the equation for XE:
Rearrange the equation and solve for XE.
You may need to make use of the quadratic formula or factorization to find the root.
By following these steps, you can find the maximum compression of the spring.