# How will we define SHM if the object is moving in uniform circular motion ?

### Asked by | 24th Mar, 2012, 09:04: AM

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alileo was the first to make observations of the moons of Jupiter. He used a telescope, at that time a recent invention, to discover four of the planets moons. He noted that the moons move in a pattern that could be best explained **if** the objects he observed were **moving** in circles around Jupiter. Using todays vocabulary, we would say that the pattern of motion he observed could be described using the concepts of simple harmonic motion. This is not to say the moons move in **SHM**, because they do not. But they appear to be doing so because an **object** **moving** in **uniform****circular** motion viewed edge-on from afar appears to be **moving** in **SHM**.

To help you understand Galileos insight, we will explain in more depth what he observed. The moons orbits are roughly **circular**. From his perspective on Earth, Galileo could only see the moons lateral motion, their motion from left to right and right to left. At the moons great distance, their motion toward and away from the Earth was not perceptible. The moons seem to repeatedly move back and forth along a straight line, as objects in **SHM** do.

When an **object** **moving** in **uniform** **circular** motion, such as a moon of Jupiter, is observed edge on, does the repeated motion actually conform to the equations for **SHM**? In Concept 1 we show a ball **moving** up and down in **circular** motion. The graph shows its vertical displacement and, as you can see, the graph is sinusoidal, like those that represent displacement in **SHM**.

In Equation 1, you can see why we perceive edge-on **uniform** **circular**motion as **SHM**. Consider the *x* displacement of the particle **moving** in**uniform** **circular** motion. The *x* displacement equals the radius (which we will call *A* here) times the cosine of the angle *?*. The angle *?* is the angular displacement. As you may recall from your studies of rotational motion, angular displacement equals the product of angular velocity and time, or*?t*. As the equation to the right reflects, the function for the *x*displacement is the same function as the one used for calculating the displacement of an **object** in **SHM**.

The relationship between **uniform** **circular** motion and **SHM** can also be confirmed qualitatively. At the perceived endpoints of its **circular** path, a moon of Jupiter would be **moving** directly away from, or toward, Galileo. In other words, its tangential velocity is directed toward or away from the Earth at these points. Here its velocity should seem to go to zero just as in **SHM**, and this is what Galileo observed.

Conversely, at the midpoint of its motion on the Earth side of Jupiter, a moon would seem to be **moving** the fastest because of the orientation of its tangential velocity. And in **SHM**, the greatest speed is observed at the midpoint of motion.

alileo was the first to make observations of the moons of Jupiter. He used a telescope, at that time a recent invention, to discover four of the planets moons. He noted that the moons move in a pattern that could be best explained **if** the objects he observed were **moving** in circles around Jupiter. Using todays vocabulary, we would say that the pattern of motion he observed could be described using the concepts of simple harmonic motion. This is not to say the moons move in **SHM**, because they do not. But they appear to be doing so because an **object** **moving** in **uniform****circular** motion viewed edge-on from afar appears to be **moving** in **SHM**.

To help you understand Galileos insight, we will explain in more depth what he observed. The moons orbits are roughly **circular**. From his perspective on Earth, Galileo could only see the moons lateral motion, their motion from left to right and right to left. At the moons great distance, their motion toward and away from the Earth was not perceptible. The moons seem to repeatedly move back and forth along a straight line, as objects in **SHM** do.

When an **object** **moving** in **uniform** **circular** motion, such as a moon of Jupiter, is observed edge on, does the repeated motion actually conform to the equations for **SHM**? In Concept 1 we show a ball **moving** up and down in **circular** motion. The graph shows its vertical displacement and, as you can see, the graph is sinusoidal, like those that represent displacement in **SHM**.

In Equation 1, you can see why we perceive edge-on **uniform** **circular**motion as **SHM**. Consider the *x* displacement of the particle **moving** in**uniform** **circular** motion. The *x* displacement equals the radius (which we will call *A* here) times the cosine of the angle *?*. The angle *?* is the angular displacement. As you may recall from your studies of rotational motion, angular displacement equals the product of angular velocity and time, or*?t*. As the equation to the right reflects, the function for the *x*displacement is the same function as the one used for calculating the displacement of an **object** in **SHM**.

The relationship between **uniform** **circular** motion and **SHM** can also be confirmed qualitatively. At the perceived endpoints of its **circular** path, a moon of Jupiter would be **moving** directly away from, or toward, Galileo. In other words, its tangential velocity is directed toward or away from the Earth at these points. Here its velocity should seem to go to zero just as in **SHM**, and this is what Galileo observed.

Conversely, at the midpoint of its motion on the Earth side of Jupiter, a moon would seem to be **moving** the fastest because of the orientation of its tangential velocity. And in **SHM**, the greatest speed is observed at the midpoint of motion.

### Answered by | 24th Mar, 2012, 10:15: AM

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