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I am having difficulty answering and understanding these theoretical questions. Any help would be greatly appreciated.

In the following physical systems that undergo simple harmonic motion, state which physical quantity or quantities you think will influence the period of motion and in what manner. Briefly justify your predictions using physical arguments rather than equations.
a. mass-spring system (oscillating horizontally on a frictionless surface)
b. simple pendulum

In an ideal mass-spring system (oscillating horizontally on a frictionless surface), the total energy is conserved. What types of energy are involved?

What is the expression for total energy
a. When the mass is at maximum displacement (x = A) in terms of A?
b. When the mass is passing through equilibrium (x = 0) in terms of v_max?

Using the fact that energy is conserved and the expressions found in the previous part, find an expression for maximum speed in terms of m, k and/or A.

Answer
For a mass on a spring increasing the mass will increase the period since more mass will accelerate more slowly as a result of Newton's 2nd Law.  Likewise, increasing the spring constant will shorten the period since a bigger spring constant produces a greater restoring force and therefore a greater acceleration.

For a mass oscillating back and forth on a frictionless horizontal surface the total energy will be the sum of the kinetic energy and the elastic potential energy:
TE=KE+EPE
Where:
TE=1/2*m*v^2+1/2*k*A^2
At maximum displacement the total energy will be:
TE=1/2*k*Amax^2  since the velocity will zero at this point
While:
TE=1/2*m*Vmax^2 since the amplitude will be zero at this point.
Making these two equal:
1/2*k*Amax^2=1/2*m*Vmax^2
Solving forthe maximum velocity:
Vmax=sqrt[k*Amax^2/m]
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I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.

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