Download An object of mass m oscillates on a vertical spring with period T. If

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An object of mass m oscillates on a vertical spring with period T. If the mass is doubled, how would the period change? A)
B)
C)
D)
The period would double
The period not change
The period would increase by a factor of SQRT(2)
The period would decrease by a factor of SQRT(2)
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Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
Also we know the position and velocity as functions of time, we can find the kinetic and potential energies:
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Energy transforms from potential to kinetic and back, while the total energy remains the same.
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A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?
a) total energy will increase
b) total energy will not change
c) total energy will decrease
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While watching a chandelier swing back and forth at the Cathedral of Pisa in 1583, Galileo noticed something curious. We might expect a chandelier swinging farther to take longer, but not so. Galileo noticed that the time period to swing through one complete cycle is independent of the amplitude through which it swings. One can duplicate Galileo's pendulum experiments by timing a weight swinging on the end of a string. For not too large amplitudes, the time period for one complete cycle will be the same regardless of amplitude. The period does however depend on the length of the string. A longer pendulum will take longer to complete one cycle. For these experiments we would use a stopwatch, perhaps one built into most cell phones these days. But in Galileo's time, wristwatches nor cell phones were not yet available. He timed the swings with his pulse, the only timing device at hand.
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The Pendulum
Looking at the forces on the pendulum mass, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
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For small angles θ (radians), sinθ is approximately equal to the angle itself. That is
The arc length displacement of the mass from equilibrium (or the vertical) is
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Note the restoring force on the pendulum has the same form If we substitute variables x = s, we can assume the equivalent k for the pendulum
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sin θ ~ θ approximation 21
A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:
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Hanging objects may be made to oscillate in a manner similar to a simple pendulum. The motion can be described by "Newton's 2nd law for rotation":
physical pendulum
simple pendulum
where the torque is
and the relevant moment of inertia is that about the point of suspension. The resulting equation of motion is:
The arc length displacement of the mass from equilibrium Rotational to linear acceleration
Multiply by mass
the equivalent k for the pendulum
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Substituting the moment of inertia of a point mass a distance L from the axis of rotation gives, as expected,
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