The block in the figure below lies on a horizontal frictionless surface and is attached to the free end of the spring, with a spring constant of 55 N/m. Initially, the spring is at its relaxed length and the block is stationary at position x = 0. Then an applied force with a constant magnitude of 2.7 N pulls the block in the positive direction of the x axis, stretching the spring until the block stops. Assume that the stopping point is reached. (a) What is the position of the block? m (b) What is the work that has been done on the block by the applied force? J (c) What is the work that has been done on the block by the spring force? J During the block's displacement, find the following values. (d) The block's position when its kinetic energy is maximum. m (e) The value of that maximum kinetic energy. mJ
I will be happy to check your thinking on this very basic problem.
To solve this problem, we can use the principles of energy conservation and Hooke's law.
(a) To find the position of the block, we need to determine how much the spring stretches in response to the applied force. We can use Hooke's law, which states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position.
The formula for Hooke's law is:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we know that the applied force has a constant magnitude of 2.7 N, and the spring constant is 55 N/m. We can set up the equation as follows:
2.7 N = -55 N/m * x
Solving for x:
x = -2.7 N / (-55 N/m)
x ≈ 0.049 m
Therefore, the position of the block is approximately 0.049 m in the positive direction of the x-axis.
(b) The work done by the applied force can be calculated using the formula:
Work = Force * Distance
In this case, the applied force has a constant magnitude of 2.7 N, and the block is displaced by a distance of 0.049 m. Substituting the values into the formula:
Work = 2.7 N * 0.049 m
Work ≈ 0.1323 J
Therefore, the work done on the block by the applied force is approximately 0.1323 J.
(c) The work done by the spring force can be calculated using the formula:
Work = (1/2) * k * x^2
In this case, the spring constant k is 55 N/m, and the block is displaced by a distance of 0.049 m. Substituting the values into the formula:
Work = (1/2) * 55 N/m * (0.049 m)^2
Work ≈ 0.0664 J
Therefore, the work done on the block by the spring force is approximately 0.0664 J.
(d) The block's position when its kinetic energy is maximum occurs when all the potential energy stored in the spring is converted into kinetic energy. This happens when the block reaches its maximum displacement from the equilibrium position.
In this case, the maximum displacement occurs at x = 0.049 m, which we found in part (a).
Therefore, the block's position when its kinetic energy is maximum is approximately 0.049 m in the positive direction of the x-axis.
(e) To find the maximum kinetic energy, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the system is conserved, which means it remains constant throughout the motion. At the maximum displacement, all the potential energy stored in the spring is converted into kinetic energy.
The potential energy stored in the spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
In this case, the spring constant k is 55 N/m, and the maximum displacement is 0.049 m. Substituting the values into the formula:
Potential Energy = (1/2) * 55 N/m * (0.049 m)^2
Potential Energy ≈ 0.0664 J
Since the total mechanical energy is conserved, the kinetic energy at the maximum displacement is equal to the potential energy at the relaxed position.
Therefore, the maximum kinetic energy is approximately 0.0664 J.