Speed in Circular Motion
Speed is equal to how far something has moved divided by how long it took to move that distance.
Distance Moved ⁄ Time
He moved one time around the circle. The distance around a circle is 2 π r. Where r is the radius of the circle
s = 2 Π r ⁄ Time
The radius of the circle is 1 meter. The time it took him to go around the circle is 2 seconds:
s = 2(3.14)(1) ⁄ 2
s = 3.14 m/s
3.14 m/s is his speed; not his velocity. Remember, speed is how fast something is moving. Velocity includes both how fast something is moving and in what direction it is moving. 3.14 m/s tells only how strong the velocity is, not what direction the velocity is in. Because he is moving in a circle, his direction is constantly changing. Nevertheless, a majority of websites and textbooks use 'v' to denote this speed even though it is not velocity. We will do the same so that we don't confuse anyone.
The Equation to find the speed of any object is:
radius of the circle: radius of the circle
speed: speed
Acceleration and Force
Acceleration in circular motion happens not because the speed is changing, but because the direction is always changing.
The acceleration an object in uniform circular motion is always directed toward center of the circle.
How storng the acceleration is depends on how fast the object is moving and how small of radius is the circle which it moves around.
Here is the equation for acceleration:
speed: speed
radius: the radius of the circle
In order to have circular motion, there must be a net force that is directed toward the center of the circle.
mass: mass
speed: speed
radius: the radius of the circle
For an object to stay in circular motion, the net force must be directed toward the center of the circle. Because the net force is always directed toward the center, so is the acceleration. The direction of the velocity, acceleration, and force are constantly changing in circular motion. Take a look at these examples:
As you can see, the force is always directed straight toward the center of the cirle. The direction of the velocity is always perpendicular to the force. Let's take a closer look at an object in circular motion to see what is happening:
The dark blue line shows the direction that the object is moving right now. The object's inertia will keep it moving in this direction unless something changes it. The red arrow shows the direction of the force on the object. If there were no force, the object would continue moving in the direciton of the blue arrow. If there were no velocity, the object would start moving in the direction of the red arrow. Because there is both velocity and force, the object will move in a direction that is somewhere between the blue arrow red arrow. The direction that the object will move is shown by the dashed green line.
Gravity
Circular motion is caused by a force that pulls an object toward the center of a circular path. Any type of force can cause circular motion. One very common cause of circular motion in the universe is the force of gravity. Gravity is responsible for the moon going around the earth, the earth going around the sun, and the sun going around galaxy.
Gravity is a force that attracts two masses together. Here is the equation for the force of gravity:
G: : 6.67 x 10-11
m1: mass of one of the objects
m2: mass of the other object
r: the distance between the center of the two masses
Looking at this equation, we can notice a few things. We notice that the force of gravity depends only on how much mass the objects involved have and how far the objects are away from each other. Playing with the numbers reveals the following information;
- The force that both masses experience is equal in strength
- If either mass is larger, gravity increases
- If the distance between masses increases, gravity quickly decreases
- If one of the masses doubles, gravity is 2x as strong
- If both of the masses double, gravity is 4x as strong
- If the distance between the two masses doubles, gravity is 1 ⁄ 4 x as strong
- If one of masses doubles and the distance doubles, gravity is 1 ⁄ 2 x as strong
- If both masses double and the distance doubles, gravity remains the same
- The strongest gravitational forces are between large masses that are close together
Gravity and Circular Motion
When one mass is much larger than another, the gravity between the two masses can be the force that causes the smaller object to orbit the larger. Remember that both masses experience an equal force. However, the larger object accelerates much less than the smaller object. This allows a situation where the smaller object's acceleration is enough to make it move in a nearly circular orbit. The larger object moves very little compared to the smaller object. We are all very familar with a couple examples of this.
The earth is much smaller than the sun. The force of gravity from the sun on to the earth is enough to cause the earth to accelerate toward the sun. The velocity that the earth already has is strong enough for the earth to escape being pulled into the sun. Likewise, the earth's gravitational force on the moon keeps the moon in orbit. The earth's gravity is enough to keep it from falling into the sun. The moon's velocity is enough to keep it from falling into the earth.
How can we know how much velocity is necessary to keep an object in orbit. We can figure this out by looking at the equation for the gravitational force and the equation for circular force:
For an object to stay in circular motion, the Force of gravity needs to equal the circular force. Let's each equation equal to each other:
Fortunately, some of these variables cancel out.
That leaves us this equation:
m2: mass of the object at the center of the system
r: the distance between the center of the two masses
v: the velocity of the object in orbit
If the velocity fits this equation, the object stays in a perfectly circular motion. A few observations about this equation:
- If the mass of the object in the center gets larger, more velocity is needed
- If distance between the objects gets farther, more velocity is needed
- If you double the mass and double the distance between the objects, the velocity can stay the same.
If the object's velocity matches the equation, it will move in a circular motion. What if the velocity is too small or too large?
If the object's velocity is too small, the object will spiral closer and closer to the center of the system.
If the object's velocity is too large, the object will move in an eliptical orbit. The faster the velocity, the more stretched out the orbit will be.
