If it is released from rest at a height of h = 2 ft from the top of the uncompressed spring, determine the speed of the collar after it falls and compresses the spring 0.9ft .
To determine the speed of the collar after it falls and compresses the spring, we can use the principle of conservation of mechanical energy.
1. First, let's find the potential energy of the collar when it is released from rest at a height of 2 ft. The potential energy is given by the formula:
Potential Energy = m × g × h
where m is the mass of the collar, g is the acceleration due to gravity (approximately 32 ft/s²), and h is the height (2 ft in this case).
2. Next, let's find the potential energy of the spring when it is compressed by 0.9 ft. The potential energy stored in a spring is given by the formula:
Potential Energy = 1/2 × k × x²
where k is the spring constant and x is the displacement of the spring (0.9 ft in this case).
3. The potential energy at the top of the uncompressed spring is equal to the potential energy at the bottom when the spring is compressed. So, we equate the potential energy of the collar (from step 1) with the potential energy of the spring (from step 2) and solve for the velocity of the collar at the bottom.
1/2 × k × x² = m × g × h
4. Once we have the velocity of the collar at the bottom, we can calculate its speed using the formula:
Speed = |velocity|
To summarize:
1. Calculate the potential energy of the collar when it is released from rest at a height of 2 ft.
2. Calculate the potential energy of the spring when it is compressed by 0.9 ft.
3. Equate the two potential energies and solve for the velocity of the collar at the bottom.
4. Calculate the speed of the collar by taking the absolute value of the velocity.
Note: To obtain accurate results, make sure to use consistent units throughout the calculation (e.g., feet and seconds).