Concept explainers
Suppose that a 300-g mass (0.30 kg) is oscillating at the end of a spring upon a horizontal surface that is essentially friction-free. The spring can be both stretched and compressed and has a spring constant of 400 N/m. It was originally stretched a distance of 12 cm (0.12 m) from its equilibrium (unstretched) position prior to release.
- a. What is its initial potential energy?
- b. What is the maximum velocity that the mass will reach in its oscillation? Where in the motion is this maximum reached?
- c. Ignoring friction, what are the values of the potential energy, kinetic energy, and velocity of the mass when the mass is 6 cm from the equilibrium position?
- d. How does the value of velocity computed in part c compare to that computed in part b? (What is the ratio of the values? This is interesting because even though you are midway between the equilibrium point and the maximum displacement, the velocity is much closer to the maximum value you found in part b.)
(a)
The initial potential energy of the spring.
Answer to Problem 4SP
The initial potential energy of the spring-mass system is
Explanation of Solution
Given info: The spring constant is
Write the expression for the elastic potential energy of a spring.
Here,
Substitute
Conclusion:
Therefore, the initial potential energy of the spring-mass system is
(b)
The maximum velocity that the mass will reach in its oscillation and the position where then system achieves maximum velocity.
Answer to Problem 4SP
The maximum velocity that the mass will reach in its oscillation is
Explanation of Solution
Given info: The mass hanged attached to the spring is
The maximum velocity will be at a point where the kinetic energy of the system is maximum. The maximum kinetic energy is obtained when all the potential energy stored in the state, is completely converted to the kinetic energy. Since the initial potential energy is
Write the expression for kinetic energy of an object.
Here,
Solve for
Substitute
The potential energy of the spring-mass system converts completely to kinetic energy when the system moves through the equilibrium position. Thus, the maximum velocity occurs as the mass moves through the equilibrium position.
Conclusion:
Therefore, the maximum velocity that the mass will reach in its oscillation is
(c)
The potential energy, kinetic energy and the velocity of the mass when the mass is
Answer to Problem 4SP
When the mass is
Explanation of Solution
Given info: The mass attached to spring is
Write the expression for the elastic potential energy of a spring.
Substitute
At a particular distance away from the equilibrium position, the system acquires some potential energy. Hence the kinetic energy at a particular position is obtained by subtracting the potential energy at that point from the maximum kinetic energy.
Thus, the kinetic energy at the given point is obtained as,
Write the expression for velocity from the expression for kinetic energy.
Substitute
Conclusion:
Therefore, when the mass is
(d)
The ratio of the values of velocity of the mass at the equilibrium position and the position
Answer to Problem 4SP
The ratio of the values of velocity of the mass at the position
Explanation of Solution
Given info: The velocity of the mass at the equilibrium position is
The ratio of the values of velocity of the mass at the position
Thus, the velocity of the mass at the position
Conclusion:
Therefore, the ratio of the values of velocity of the mass at the position
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