a pendulum consisting of a sphere suspended from a light string is oscillating with a small angle with respect to the vertical. the sphere is then replaced with a new sphere of the same size but greater density and is set into oscillation with the same angle. how do the period, maximum kinetic energy, and maximum acceleration of the new pendulum compare to those of the original pendulum?
How do the period, maximum kinetic energy, and maximum acceleration of the new pendulum compare to those of the original pendulum?
Answer:
When the sphere of the original pendulum is replaced with a new sphere of the same size but greater density, several key properties will change:
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Period: The period of the pendulum (the time it takes to complete one full oscillation) does not depend on the mass of the pendulum bob. Therefore, the period of the new pendulum with the denser sphere will be the same as the period of the original pendulum.
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Maximum Kinetic Energy: Since kinetic energy is directly proportional to the square of the velocity and the velocity is maximum at the equilibrium position (lowest point of the swing), the new denser sphere will have a higher maximum kinetic energy compared to the original pendulum due to the greater density of the sphere and thus, higher mass.
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Maximum Acceleration: The maximum acceleration of a pendulum is also independent of the mass of the pendulum bob, as it is determined by the length of the string and the angle of displacement. Therefore, the maximum acceleration of the new pendulum with the denser sphere will be the same as the original pendulum.
In summary, the period and maximum acceleration of the new pendulum will remain the same as that of the original pendulum, while the maximum kinetic energy of the new pendulum will be higher due to the increased density of the sphere.