Concept explainers
Review. A particle of mass 4.00 kg is attached to a spring with a force constant of 100 N/m. It is oscillating on a frictionless, horizontal surface with an amplitude of 2.00 m. A 6.00-kg object is dropped vertically on top of the 4.00-kg object as it passes through its equilibrium point. The two objects stick together. (a) What is the new amplitude of the vibrating system after the collision? (b) By what factor has the period of the system changed? (c) By how much does the energy of the system change as a result of the collision? (d) Account for the change in energy.
(a)
The new amplitude of the vibration system after collision.
Answer to Problem 37AP
The new amplitude of the vibration system after collision is
Explanation of Solution
Section 1:
To determine: The angular frequency of the system.
Answer: The angular frequency of the system is
Given information: The mass of the particle is
The formula for the angular frequency is,
Substitute
Section 2:
To determine: The maximum speed of the system.
Answer: The maximum speed of the system is
Given information: The mass of the particle is
The formula to calculate maximum speed is,
Substitute
Section 3:
To determine: The speed of the system when the objects stick together after the collision.
Answer: The speed of the system when the objects stick together after the collision is
Given information: The mass of the particle is
The formula to calculate speed after the collision is,
Substitute
Section 4:
To determine: The new amplitude of the vibration system after collision.
Answer: The new amplitude of the vibration system after collision is
Given info: The mass of the particle is
The law of conservation of energy is,
Rearrange the above equation for
Substitute
Conclusion:
Therefore, the new amplitude of the vibration system after collision is
(b)
The factor by which the period of system changed.
Answer to Problem 37AP
The factor by which the period of system changed is
Explanation of Solution
Section 1:
To determine: The initial period of system.
Answer: The initial period of system is
Given info: The mass of the particle is
The formula for the period of the system before collision is,
Substitute
Section 2:
To determine: The final period of system.
Answer: The final period of system is
Given info: The mass of the particle is
The formula for the period of the system after collision is,
Substitute
Section 3:
To determine: The factor by which the period of system changed.
Answer: The factor by which the period of system changed is
Given info: The mass of the particle is
The factor by which period is changed calculated as,
Substitute
Conclusion:
Therefore, the factor by which the period of system changed is
(c)
The energy changed of the system after the collision.
Answer to Problem 37AP
The energy of the system after the collision is decreased by factor
Explanation of Solution
Given info: The mass of the particle is
The formula for the energy of the system before collision is,
The formula for the energy of the system after collision is,
The chance in the energy is calculated as,
Substitute
Substitute
Conclusion:
Therefore, the energy of the system after the collision is decreased by factor
(d)
To explain: The change in the energy.
Explanation of Solution
The energy of the system is defined as the capacity to do any work. The energy is the sum of potential and the kinetic energy of the system.
The type of the collision of the system is inelastic due to this the kinetic energy does not remains conserved. The mechanical energy of the system is transformed into the internal energy. So there are energy losses due to conversion of energy.
Conclusion:
Therefore, the mechanical energy of the system is transformed into the internal energy in the perfectly inelastic collision.
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