Concept explainers
A particle with a mass of 0.500 kg is attached to a horizontal spring with a force constant of 50.0 N/m. At the moment t = 0, the particle has its maximum speed of 20.0 m/s and is moving to the left. (a) Determine the particle’s equation of motion, specifying its position as a function of time. (b) Where in the motion is the potential energy three times the kinetic energy? (c) Find the minimum time interval required for the particle to move from x = 0 to x = 1.0 m. (d) Find the length of a simple pendulum with the same period.
(a)
The equation of the motion of the particle.
Answer to Problem 15.79AP
The motion equation of the motion of the particle is
Explanation of Solution
Mass of the particle is
Formula to calculate the angular frequency is,
The general equation of the particle’s motion is,
Differentiate the above equation with respect to time.
From the given condition, At
Substitute these values in the above equation.
The maximum value of
Therefore,
Substitute
Substitute
Substitute
Here,
Conclusion:
Therefore, the motion equation of the motion of the particle is
(b)
The position where the potential energy is the three times the kinetic energy.
Answer to Problem 15.79AP
The position where the potential energy is the three times the kinetic energy is
Explanation of Solution
Mass of the particle is
Formula to calculate the maximum energy stored in the spring is,
Formula to calculate the potential energy at any position is,
Formula to calculate the kinetic energy at any position is,
From the given condition,
From the conservation of energy,
Substitute
Substitute
Conclusion:
Therefore, the position where the potential energy is the three times the kinetic energy is
(c)
The minimum time interval required for the particle to move from
Answer to Problem 15.79AP
The minimum time interval required for the particle to move from
Explanation of Solution
Mass of the particle is
The position of the particle is given by,
So, the particle will be at
Initially, at
So, the particle will be at
Therefore, the minimum time interval required for the particle to move from
Conclusion:
Therefore, the minimum time interval required for the particle to move from
(d)
The length of a simple pendulum with same period.
Answer to Problem 15.79AP
The length of a simple pendulum with same period is
Explanation of Solution
Formula to calculate the length of the pendulum is,
Here,
Substitute
Conclusion:
Therefore, the length of a simple pendulum with same period is
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Chapter 15 Solutions
PHYSICS:F/SCI.+.,V.2-STUD.S.M.+STD.GDE.
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