If a 40 kg brick and a 400 kg brick each dropped from 1 m above a trampoline, find the lowest position of each brick. Assume the trampoline is a simple spring obeying Hooke's law with a k value of 12 000 N/m.
So, I used
mg(h+x)=1/2 k x^2
then solved for x in each case.
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Bob, could you please tell me how you came up with this eq'n? Like, what evergies you started with?
Potential energy of the brick lost as it fell a height from the trampoline and deformed it x meters.
mg(h+x).
2) Potential energy stored in the deformed trampoline. 1/2 k (x^2)
set them equal.
Sure! To derive the equation you used, let's start by considering the different energies involved.
First, we have the gravitational potential energy (PE) associated with each brick. The potential energy is given by the equation PE = mgh, where m is the mass of the brick, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the brick is dropped.
When the brick hits the trampoline, it compresses the spring, storing potential energy in the spring. This potential energy can be calculated using Hooke's law. Hooke's law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as F = kx, where F is the force, k is the spring constant (k value of 12,000 N/m in this case), and x is the displacement.
Now, in the case of the brick on the trampoline, the gravitational potential energy is converted into potential energy stored in the spring when it compresses. At the lowest position, all the gravitational potential energy will be converted into potential energy of the spring. Thus, we can equate the two energies.
mg(h + x) = 1/2 kx²
Here, m is the mass of the brick, g is the acceleration due to gravity, h is the initial height from which the brick is dropped, x is the displacement of the spring from its equilibrium position, and k is the spring constant.
By rearranging this equation and solving for x, we can find the lowest position (displacement) of each brick when dropped from 1 m above the trampoline.
I hope that clarifies the derivation of the equation for you! Let me know if you have any further questions.