An 6.82 kg block drops straight down from a height of 1.31 m, striking a platform spring having a force constant of 1.16 10^3 N/m. Find the maximum compression of the spring.
To find the maximum compression of the spring, you need to use the principle of conservation of energy. Here's how you can do it step by step:
1. First, calculate the potential energy of the block when it is at the top of the platform (before it falls). The potential energy is given by the formula: PE = mgh, where m is the mass of the block (6.82 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (1.31 m).
PE = (6.82 kg) × (9.8 m/s^2) × (1.31 m)
PE ≈ 88.728 J
2. The potential energy is converted into kinetic energy as the block falls. The kinetic energy is given by the formula: KE = 1/2 mv^2, where m is the mass of the block (6.82 kg) and v is the velocity of the block when it hits the spring.
3. At the bottom of the fall, just before the block hits the spring, all of the potential energy is converted into kinetic energy. Therefore, the kinetic energy equals the potential energy: KE = PE.
1/2 mv^2 = 88.728 J
4. Now, we can solve for the velocity (v) of the block just before it hits the spring. Rearranging the equation, we have:
v^2 = (2 × 88.728 J) / 6.82 kg
v^2 ≈ 25.99 m^2/s^2
Taking the square root, we get:
v ≈ 5.1 m/s
5. Finally, we can calculate the compression of the spring using Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. The formula for the force exerted by the spring is: F = kx, where k is the force constant of the spring (1.16 × 10^3 N/m) and x is the compression of the spring.
The force acting on the block just before it hits the spring is equal to its weight: F = mg = (6.82 kg) × (9.8 m/s^2) ≈ 66.916 N
Since the force exerted by the spring is equal to the weight of the block, we have:
kx = mg
(1.16 × 10^3 N/m) × x = 66.916 N
Solving for x, we find:
x ≈ 0.058 m
Therefore, the maximum compression of the spring is approximately 0.058 meters.
gravitational potential energy -->kinetic energy of block ---> potential energy of spring
m g h = (1/2) m v^2 = (1/2) k x^2