wo blocks with masses m1 = 1.4 kg and m2 = 3.6 kg are at rest on a frictionless surface with a compressed spring between them. The spring is initially compressed by 60.0 cm and has negligible mass. When both blocks are released simultaneously and the spring has dropped to the surface, m1 is found to have a speed of 2.4 m/s.What is the spring constant of the spring?
To find the spring constant of the spring, we can use the principle of conservation of mechanical energy. We know that the initial potential energy stored in the compressed spring is equal to the final kinetic energy of both blocks.
1. First, let's calculate the initial potential energy stored in the spring. The potential energy stored in a spring is given by the equation: PE = (1/2)kx^2, where k is the spring constant and x is the displacement of the spring from its equilibrium position (in this case, the compression).
Given that the spring is compressed by 60.0 cm (or 0.60 m), we can calculate the initial potential energy as follows:
PE_initial = (1/2) * k * (0.60)^2
2. Next, let's calculate the final kinetic energy of block m1. The kinetic energy of an object is given by the equation: KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.
Given that block m1 has a speed of 2.4 m/s, we can calculate its final kinetic energy as follows:
KE_final = (1/2) * m1 * (2.4)^2
3. Since the spring is initially compressed and both blocks are at rest, the total initial kinetic energy is zero.
Therefore, we can write the conservation of mechanical energy equation as:
PE_initial + 0 = KE_final
Now, we can substitute the values into the equation and solve for the spring constant (k).
(1/2) * k * (0.60)^2 = (1/2) * m1 * (2.4)^2
4. Rearranging the equation, we get:
k = (m1 * (2.4)^2) / (0.60)^2
Substituting the given values of m1 = 1.4 kg and solving the equation, we get:
k = (1.4 * (2.4)^2) / (0.60)^2
Calculating the expression yields the spring constant, k = 49.28 N/m.
Therefore, the spring constant of the spring is 49.28 N/m.