(a)
The initial potential energy.
(a)
Answer to Problem 4SP
The initial potential energy is
Explanation of Solution
Given info: The spring constant is
Write the expression for the elastic potential energy of a spring.
Here,
Substitute
Conclusion:
Thus 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.
(b)
Answer to Problem 4SP
The maximum velocity that the mass will reach in its oscillation is
Explanation of Solution
Given info: The mass 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,
Rewrite the above equation for
Substitute
The potential energy of the spring-mass system converts completely to kinetic energy when the system moves through the equilibrium position. This implies 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
(c)
Answer to Problem 4SP
When the mass is
Explanation of Solution
Given info: The mass attached to spring is
Substitute
At equilibrium position total energy is equal to the kinetic energy of the system. But as the system moves from the equilibrium position, it gains potential energy and there will be decrease in kinetic energy since total energy is constant. Hence the kinetic energy at a particular point 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,
Substitute
Conclusion:
Thus when the mass is
(d)
The comparison of velocity of the mass at the equilibrium position to the value when it is
(d)
Answer to Problem 4SP
The velocity of the mass when it is
Explanation of Solution
Take the ratio of the values of velocity of the mass when it is
Here,
Substitute
Thus, the velocity of the mass at the position
Conclusion:
Thus the velocity of the mass when it is
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