A 20.0 kg block is connected to a 30.0 kg block by a string that passes over a light, frictionless pulley. The 30.0 kg block is connected to a spring that has negligible mass and a force constant of 300 N/m, as shown in the figure below. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0 kg block is pulled 20.0 cm down the incline (so that the 30.0 kg block is 40.0 cm above the floor) and released from rest. Find the speed of each block when the 30.0 kg block is 20.0 cm above the floor (that is, when the spring is unstretched).
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To find the speed of each block when the 30.0 kg block is 20.0 cm above the floor (when the spring is unstretched), we need to use concepts of energy and Newton's laws of motion.
First, let's determine the potential energy and the spring potential energy at the initial position and the final position.
The potential energy on the inclined plane can be calculated using the formula:
Potential Energy = mass * gravitational acceleration * height
For the 20.0 kg block:
Potential Energy_initial = 20.0 kg * 9.8 m/s^2 * 0.20 m = 39.2 J
Potential Energy_final = 20.0 kg * 9.8 m/s^2 * 0.0 m = 0 J
For the 30.0 kg block:
Potential Energy_initial = 30.0 kg * 9.8 m/s^2 * 0.40 m = 117.6 J
Potential Energy_final = 30.0 kg * 9.8 m/s^2 * 0.20 m = 58.8 J
Next, we calculate the spring potential energy:
Spring Potential Energy_initial = (1/2) * spring constant * (extension of the spring)^2
Spring Potential Energy_initial = (1/2) * 300 N/m * (0.40 m - 0.20 m)^2 = 6 J
Spring Potential Energy_final = (1/2) * 300 N/m * (0.20 m - 0.20 m)^2 = 0 J
The total initial mechanical energy of the system is the sum of the initial potential energy and the initial spring potential energy:
Total Initial Energy = Potential Energy_initial + Spring Potential Energy_initial
Total Initial Energy = 39.2 J + 117.6 J + 6 J = 162.8 J
The total final mechanical energy of the system is the sum of the final potential energy and the final spring potential energy:
Total Final Energy = Potential Energy_final + Spring Potential Energy_final
Total Final Energy = 0 J + 58.8 J + 0 J = 58.8 J
According to the law of conservation of energy, the total mechanical energy of a system remains constant if no external forces act on it. Therefore, the total initial energy is equal to the total final energy:
Total Initial Energy = Total Final Energy
162.8 J = 58.8 J + Kinetic Energy_final
Now, we can solve for the final kinetic energy:
Kinetic Energy_final = Total Initial Energy - Total Final Energy
Kinetic Energy_final = 162.8 J - 58.8 J = 104 J
Since both blocks are connected by a string and move together, they will have the same final velocity.
The kinetic energy of each block is given by the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Solving for velocity:
104 J = (1/2) * (20.0 kg + 30.0 kg) * velocity^2
104 J = (1/2) * 50.0 kg * velocity^2
velocity^2 = (104 J * 2) / 50.0 kg
velocity^2 = 4.16 m^2/s^2
Taking the square root of both sides:
velocity = √(4.16 m^2/s^2)
velocity ≈ 2.04 m/s
Therefore, the speed of each block when the 30.0 kg block is 20.0 cm above the floor is approximately 2.04 m/s.