A 1.80 kg block slides on a frictionless horizontal surface. The block hits a spring with a speed of 2.00 m/s and compresses it a distance of 11.0 cm before coming to rest. What is the force constant of the spring?
To find the force constant of the spring, you can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as F = -kx, where F is the force exerted by the spring, k is the force constant, and x is the displacement from the equilibrium position.
Given the mass of the block (m = 1.80 kg) and the speed at which it hits the spring (v = 2.00 m/s), we can calculate the kinetic energy of the block just before it hits the spring using the formula KE = 1/2mv^2.
KE = 1/2 * 1.80 kg * (2.00 m/s)^2
KE = 3.60 J
The kinetic energy of the block will be converted into potential energy stored in the compressed spring. Since the block comes to rest, all of its initial kinetic energy is used to compress the spring. Thus, the potential energy (PE) stored in the spring can be calculated using the formula PE = 1/2kx^2.
PE = 1/2 * k * (0.11 m)^2
PE = 0.0055 k J
Equating the kinetic energy to the potential energy, we have:
3.60 J = 0.0055 k J
Dividing both sides of the equation by 0.0055 J to isolate k, we get:
k = 3.60 J / 0.0055 J
k ≈ 654.55 N/m
Therefore, the force constant of the spring is approximately 654.55 N/m.