HOLT text reference: Chapter 12

Your textbook links the concept of simple harmonicmotion (SHM) with the introduction to waves so we will as well.

SHM is the result of periodic (pattern centered and repeated) motion that is the result of a restorative force which is proportional to the displacement of an object from a position of equilibrium. Not all periodic motion is SHM.

How to determine if a periodic motion represents SHM:

1. Motion that is not SHM: Suppose you had a row of equally spaced lights that are designed to turn on for a set amount of time and then off again in a pattern from one end to the other and than back again. From a distance, the lights appears to be moving from left to right and then from right to left at a constant speed. In SHM the speed is not constant so this is not SHM simply based on this factor.

2. Motion that is SHM: Suppose you were swimming at a favorite spot along a river that had a overhanging tree with a rope swing attached. You climb to a high bank overlooking the river and swing out over the water and let go. Had you simply hung on and allowed the rope to swing back and forth it would eventually stop with you holding the rope such that you and the rope were perpendicular to the surface of the water. This position is the point of equilibrium and the bank you jumped from represented the maximum displacement from that equilibrium location. This scenario represents SHM because your motion would not have been at a constant rate when you jumped from the bank. The "restorative force" involved is gravity. That force positively accelerates you as you move toward the point of equilibrium and negatively accelerates you as you move away from that point. As you moved away from the point of equilibrium on the other side, eventually you would stop and the process would repeat itself as you came back down.

Hooke's Law:

Hooke's Law describes the relationship between a spring that is stretched and SHM. It states that the elastic force (F_elastic) generated by a spring is the product of the negative factor of that spring's spring constant (-k) and the amount of displacement (x in meters) that produces that force. Both are directly proportional to the elastic force generated.

where: F is measured in N, k is measured in N/m, and x is measured in meters

NOTE: The negative sign simply means that the direction of the force elastic is opposite to the direction of the displacement of the spring.

Springs are not always oriented in such a way as to eliminate the force of gravity.

Turn in your textbook to page 440. Go through example as presented.

Alternative approach:

Since the weight (-mg) of the object pulls the spring downward...

The spring stretches until its restoring force (F_elastic = -kx) balances the -5.4 N. This occurs when x = -.20 m. This means that...

Try these:

1. A 76 N crate is attached to spring (k= 450 N/m). How much displacement is caused by the weight of the crate? (Answer: -.17 m)

2. A spring (k = 1962 N/m) loses its elasticity if stretched more than 50.0 cm. What is the mass of the heaviest object the spring can support without being damaged?

The Simple Pendulum:

The river bank example of SHM is also an example of a device called a simple pendulum. It has the following defining characteristics:

1. Amplitude: This the maximum displacement from the point of equilibrium.

2. Period (T): This is the time (s) required to make 1 complete back and forth cycle of SHM motion. Starting point to starting point.

3. Frequency (f): This is the number of cycle per unit time making the frequency the reciprocal of the period (T = 1/f and f = 1/T). The SI unit for f is the Hertz (Hz) and since f = 1/T, s^-1 = Hz as well.

Calculating the period (T) of a SHM in a pendulum requires a new equation:

This equation is rather revealing with regard to what does and what does not affect the period of a pendulum:

1. Factors that do include the length of the pendulum arm and the acceleration due to gravity.

2. Factors that do not include the mass of the object being swung on the pendulum and the amplitude (for amplitude angles of <15^o) .

Turn to page 448 in your textbook and look at example 12B. Go through this example.

Now try these:

1. What is the period of a 3.98 m and a 99.4 cm pendulum?

ans. 4.00 s and 2.00 s respectively

2. A desktop toy swings back and forth once each second. How tall is this toy?

ans. .25 m

3. What is the free-fall acceleration at a location where a 6.00 m long pendulum swings through exactly 100 cycles in 492 s?

ans. 9.79 m/s²

What affects the period of a mass-spring system?

1. The mass of the object being displaced.

2. The value of the spring constant because it is the spring constant that determines how much force is required to displace the mass per unit distance.

The period for a mass-spring system therefore has a distinctively different equation to consider.

Now consider example 12C on page 450. Go through this example too.

Now try these:

1. A 1.0 kg mass attached to one end of a spring completes an oscillation once every 2.0 s. Find the spring constant.

ans. 9.9 N/m

2. What size mass will make this spring vibrate once each s?

ans. .25 kg

Assignment: Do practice sessions 12 A - 12 C
 

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