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 speed of m2?
To find the speed of m2, we can start by applying the principle of conservation of mechanical energy. When the blocks are released, the potential energy stored in the compressed spring is converted into the kinetic energy of the blocks.
We can start by calculating the potential energy stored in the spring when it was compressed. The equation for potential energy of a spring is given by:
Potential energy (PE) = (1/2) * k * x^2
Where k is the spring constant and x is the displacement or compression of the spring. In this case, the compression of the spring is given as 60.0 cm, which is equal to 0.60 m.
Next, let's consider the kinetic energy of block m1 when it has a speed of 2.4 m/s. The equation for kinetic energy is given by:
Kinetic energy (KE) = (1/2) * m * v^2
Where m is the mass of the block and v is its speed.
Since block m1 is at rest initially, its kinetic energy is zero. So, all the initial potential energy stored in the spring is converted into the kinetic energy of block m1.
Setting the potential energy of the spring equal to the kinetic energy of block m1, we have:
(1/2) * k * x^2 = (1/2) * m1 * v1^2
Substituting the given values, we can solve for the spring constant, k:
(1/2) * k * (0.60)^2 = (1/2) * 1.4 * (2.4)^2
Simplifying the equation, we have:
k * 0.36 = 3.36
k = 3.36 / 0.36
k ≈ 9.33 N/m
Now that we have the spring constant, we can find the speed of block m2. Since the total mechanical energy is conserved, the sum of the potential energy and kinetic energy after the spring has dropped to the surface is equal to the initial potential energy stored in the compressed spring.
The final potential energy after the spring has dropped is zero since it is no longer compressed. Therefore, the final kinetic energy of block m1 is equal to the initial potential energy of the spring.
(1/2) * m1 * v1^2 = (1/2) * k * x^2
Substituting the given values and solving for the speed of m2, we have:
(1/2) * 1.4 * (2.4)^2 = (1/2) * 9.33 * 0.60^2 + (1/2) * 3.6 * v2^2
(1/2) * 1.4 * (2.4)^2 - (1/2) * 9.33 * 0.60^2 = (1/2) * 3.6 * v2^2
Simplifying the equation, we have:
8.064 - 1.065 = 1.8 * v2^2
6.999 = 1.8 * v2^2
v2^2 ≈ 3.888
Taking the square root of both sides, we find:
v2 ≈ 1.972 m/s
Therefore, the speed of m2 is approximately 1.972 m/s.