A block of mass m = 2.4 kg is released from rest at a height of H = 3.0 m on a curved frictionless ramp. At the foot of the ramp is a spring whose spring constant is k = 317.0 N/m. What is the maximum compression of the spring, x?
equate PE=spring potential energy (1/2)kx²
To find the maximum compression of the spring, we need to consider the conservation of mechanical energy for the system.
The initial potential energy of the block at height H is converted into the final potential energy of the block-spring system when the block reaches its maximum compression.
Let's go step by step to find the solution:
Step 1: Calculate the initial potential energy (PE_initial) of the block at height H.
The potential energy (PE) is given by the formula PE = m * g * h, where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
PE_initial = m * g * H
Substituting the given values, we can calculate PE_initial.
Step 2: Calculate the final potential energy (PE_final) of the block-spring system.
At maximum compression, all the initial potential energy is converted into potential energy stored in the spring.
PE_final = (1/2) * k * x^2
where k is the spring constant and x is the maximum compression.
Step 3: Equate the initial and final potential energies and solve for x.
PE_initial = PE_final
m * g * H = (1/2) * k * x^2
Substituting the given values, we can calculate x.
Note: Make sure all the units are consistent (such as meters for height and kilograms for mass) to get the correct answer.
By following these steps, you can find the maximum compression of the spring in the given scenario.