Horizontal spring in simple harmonic motion.

1. Physics 101: Learning Object 1
Horizontal Mass Spring System Question
A block of mass 0.74kg sits on a horizontal frictionless surface attached
to a spring. The mass oscillates in simple harmonic motion with a period
of 0.016 minutes. When the mass reaches 15.20cm from its equilibrium
position its speed is 3.25m/s.
a) What is the value of the spring constant?
b) What is the maximum distance that the block can be from its
equilibrium position?

2. SOLUTION
a) To solve for the spring constant we can use the period. First we must convert the period given to seconds.
Now using the formula we can solve for k, also known as the spring constant.
Simply substitute in the values and find k.
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b) The important thing to know is that because the block is in simple harmonic motion on a horizontal frictionless surface, the
height is 0 and there is no gravitational potential energy. Knowing this, the total energy of the block will be
Which is the total kinetic energy of block plus the elastic potential energy of the spring.
Now to find the the furthest distance the block can reach, or the amplitude, we realize that the velocity will be equal to zero because
all the kinetic energy is stored in the potential energy of the spring as a result of energy conservation in simple harmonic motion. So
now our equation is
The maximum distance is equal to the amplitude so we can replace x (the distance) with A in the formula, as the distance we want in
this case is the amplitude.
A picture will help our understanding of what we just found.
Now that we have our formula for finding the amplitude, or the maximum distance, we need to solve for the other unknown variable
which is the total energy of the system. So now going back to the formula we can solve for E by substituting in the
values given from the question and the spring constant value we solved in part a.
Now going back to the formula we found prior we can solve for the amplitude.