If you have used a GAPS, or even a mobile phone or satellite TV or radio, you have used mom of the nearly 300 artificial satellites that exist above our atmosphere to send and receive information. Satellites are projectiles that, through our knowledge of force and energy, have been launched into orbit. Because the earth curves downward by approximately 5 meters over 8000 meters along the horizon, satellites launched with a horizontal speed of 8000 m/s can take orbit.

At this speed, thanks to earth’s curved surface and a satellite’s constant horizontal velocity, the satellites, which are in free fall once projected, keep falling toward the earth, but always miss it. Thus, under the influence of gravity, satellites maintain motion in a circular pattern at a uniform speed. Most satellites are launched using rockets that fall to the ocean when their fuel is spent. Sometimes a satellite may require minor adjustments to its orbit. This is accomplished through the use of built-in rockets, called thrusters.

Once placed in the proper orbit, a satellite can stay there for a long time, with its operation and location monitored by computers and human operators at a control centre on earth. Solar panels provide a source of power for some satellites. A satellite keeps its solar panels facing the sun and its antennae ready to receive information. A satellite remains in orbit until its speed decreases and gravity pulls it down into the atmosphere, where it slows down, due to its collisions with air molecules in the atmosphere.

As the satellite falls further down into the denser atmosphere, it compresses the air in front of it, causing the air to become so hot that the satellite burns up. In this lesson, you will learn more about the forces that contribute to the motion of a satellite and to other circular motions. Planning Your Study You may find this time grid helpful in planning when and how you will work through Suggested Timing for This Lesson (hours) Investigation into Circular Motion Problem Solving with Circular Motion Frames of Reference Key Questions LLC. Erg Copyright 0 2011 The Ontario Educational Communications Authority. All rights reserved. 2 What You Will Learn After completing this lesson, you will be able to describe the relationships between variables affecting uniform circular motion identify and analyze the forces contributing to uniform circular motion seditious (apparent) forces Copyright 2011 The Ontario Educational Communications Authority. All rights Before you begin this lesson, find an object that you can safely tie to a string and swing around in a circle, for example, a you-you, or even a belt with a heavy buckle.

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A heavier object at the end of the rope will give you a better “feel” for the ideas in this lesson, but safety should be your primary concern. Make sure that you have lots of space in which to swing the object and put on some safety glasses or sports goggles, if you have them. Try to swing your object so that it stays on a constant horizontal circular path (as if you were getting ready to lasso an animal). After this, try to make a vertical circular motion. Notice the differences.

Can you feel how the tug on your hand changes at different times, in the vertical cycle? That’s what this lesson is all about: the forces involved in creating uniform circular motion. Uniform Circular Motion Uniform circular motion is motion in which an object travels in a circle, maintaining a constant radius and constant speed. The following diagram shows the path of an abject in uniform circular motion and indicates three specific points in its path. The vectors indicate the object’s instantaneous velocity.

Notice that the length of each vector is the same, indicating that the magnitude of velocity at all locations is the same. The direction of the vectors indicates the direction of the instantaneous velocity at each location, that is, the direction in which the object would travel, if the string to which it is connected suddenly snapped. Therefore, even though the magnitude of the velocity of the ball is not changing, the direction of the velocity is instantly changing, indicating that there is uniform acceleration, despite the constant speed. Ђ?+ 3 Centripetal Acceleration The type of acceleration that results in uniform circular motion is called centripetal acceleration ( AC ), also known as radial acceleration (because it is directed along the radius of the path of motion). Centripetal acceleration is always directed toward the centre of the circular path created by the motion; therefore, it is always perpendicular to the direction of the instantaneous velocity. Consider an object going around a circle with a constant speed and a constant Addis.

If this object, at this particular location in the circular path, had continued to travel in a straight line instead of being forced to maintain a circular path, then, rearranging the basic equation for determining velocity, the displacement would have been equal to A d = v Ad However, the displacement was not equal to v At because the object experienced acceleration in -?+ Ad -? order to stay on the circle. As a result, the object’s displacement is actually equal to v At, plus the displacement that occurred due to the uniform acceleration: AC At 2 .

This is eased on the 2 kinematics equation you know for finding the displacement of an object that is experiencing -?+ uniform acceleration: A d = vi At + a At 2. Notice that, in the diagram, the direction of the 2 displacement due to acceleration is in the same direction as the centripetal acceleration: toward the centre of the circle. Ad = v At x = AC At 2 By drawing in the various displacement vectors for this object, a right triangle is formed. Notice that the displacement due to the acceleration is being set as equal to x.

Applying the Pythagorean theorem: Given that A d v At , the equation can be rewritten as: Expanding and simplifying, you get: r 2+vat=re+arc+xx vat=arc+xx Look at the diagram again. If the two object locations that had been chosen were much closer together, what would happen to x? Can you see that as the two points get closer, x will become 2 smaller? In fact, as the time interval of the change in position approaches zero, x will approach zero faster, becoming negligible. This actually involves a little bit of calculus. ) *2 Given that=becomes negligible, you can eliminate x to get: r 2 + v At r 2+arc+xx Solving for x, you get: v At 2 = arc 5 6 Recall that: Substituting this into the equation and simplifying, you get an equation for centripetal acceleration: vat AC At = In the equations above, you have written the vectors as they are meant to be communicated, using arrows above the variables.

While this is correct, it is a common convention, when working with uniform circular motion, to omit the arrows above the vector symbols, even though you are talking about vector quantities. This simply means that the magnitude of the vectors-?not the direction-?is being considered in he equations. This is an acceptable convention in uniform circular motion questions, since the direction of the instantaneous velocity is always tangent to the circle. In previous math courses, you learned that the circumference of a circle is equal to nor.

Similarly, you may recall that the time required to complete one rotation is referred to as the period of revolution and is given the symbol T. Since the magnitude of velocity of an object is Ad , the magnitude of velocity in uniform circular motion can be calculated through the At nor equation: v = Substituting this expression into the centripetal acceleration equation you have derived gives you: try TOT AC = 2 where f is the frequency of rotation (that is, how often the f rotations happen, measured in hertz).

Therefore, if you know the frequency (f) in a situation, you may use the equation: You may also recall that T = Equation Summary You now have three equations for calculating centripetal acceleration: 2 1. AC AC = r 2. AC 3. AC = 411 ref So, if an object is in uniform circular motion, you have the tools to analyze that motion. However, as you learned in the last lesson, where there is acceleration, there must be force. What causes uniform circular motion?

According to Newton’s second law, whenever there is acceleration, there must be a net force causing that acceleration. The net force that keeps an object moving in uniform circular motion is referred to as centripetal force. Centripetal force is not a new “type” of force; it is simply the sum of the combination of forces that contribute to the circular motion. In other words, it is Just another name for net force in a situation where there is uniform circular motion.

Regardless of which forces contribute to the centripetal force causing the object to follow a circular path, the direction of this net force is always aimed toward the centre of the circle, in the same direction as the centripetal acceleration. Example 1 In this diagram of a ball that is rotating clockwise in a horizontal plane, determine the directions of the velocity, centripetal acceleration, and centripetal force at the location indicated. W s 7 8 Solution The velocity is directed south (tangent to the circle and in the direction in which the ball would move if the string broke, at this instant).


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