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**Chapter 7: Circular Motion and Gravitation**

Coach Kelsoe Physics Pages 233–267

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**Section 7–1: Circular Motion**

Coach Kelsoe Physics Pages 234–239

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Section 7–1 Objectives: Solve problems involving centripetal acceleration. Solve problems involving centripetal force. Explain how the apparent existence of an outward force in circular motion can be explained as inertia resisting the centripetal force.

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**Centripetal Acceleration**

The cars on a rotating Ferris wheel are said to be in circular motion. Any object that revolves about a single axis undergoes circular motion.

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Tangential Speed The line about which the rotation occurs is called the axis of rotation. Tangential speed (vt) can be used to describe the speed of an object in circular motion.

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Tangential Speed The tangential speed of a car on the Ferris wheel is the car’s speed along an imaginary line drawn tangent to the car’s circular path. This definition can be applied to any object moving in circular motion. When the tangential speed is constant, the motion is described as uniform circular motion.

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Tangential Speed Tangential speed depends on the distance from the object to the center of the circular path. To understand this concept, imagine a carousel. The horses or carts on a carousel are staggered so that some are on the outside edge while some are closer to the middle.

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Tangential Speed Each horse/cart completes one circle in the same time period, but the outside ones cover more area, therefore must have a greater tangential speed.

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**Centripetal Acceleration**

If the cart on a Ferris wheel is moving at a constant speed around the wheel, it still has an acceleration. Even though we typically think of acceleration being a change of speed, it can also be a change of direction. On a Ferris wheel, the carts are constantly changing direction.

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**Centripetal Acceleration**

An acceleration of this nature is called a centripetal acceleration – the acceleration directed toward the center of a circular path. The equation for finding the magnitude of centripetal acceleration is given below: ac = vt2/r

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**Centripetal Acceleration**

Since acceleration is a vector quantity, we need to know the direction of the acceleration. But if direction constantly changes, how can we accurately define the direction? Centripetal acceleration is always toward the center of the circle! The word “centripetal” actually means “center seeking.” We can better understand this idea by drawing tangent vector lines or by adding opposite vectors at two points.

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Sample Problem A A test car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s2, what is the car’s tangential speed?

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**Sample Problem A Given: Unknown Solve r = 48.2 m ac = 8.05 m/s2 vt = ?**

ac = vt2/r , so… vt = √acr = √(8.05 m/s2)(48.2 m) vt = 19.7 m/s

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**Tangential Acceleration**

Centripetal acceleration results from a change in direction, not a change in speed. In circular motion, an acceleration due to a change in speed is called tangential acceleration. The easiest way to think of this is a car on a circular track – it has centripetal acceleration no matter what, due to its change in direction. It will only have tangential acceleration if it speeds up or slows down.

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Centripetal Force Consider a ball of mass m that is tied to a string of a length r and that is being whirled in a horizontal circular path. Assume the ball moves with a constant speed. r m

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Centripetal Force Assume that the ball moves with constant speed. Because the velocity vector, v, continuously changes direction during the motion, the ball experiences a centripetal acceleration that is directed toward the center of the motion. The inertia of the ball tends to maintain the ball’s motion in a straight path. However, the string exerts a force that overcomes this tendency. r m

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Centripetal Force The net force that is directed toward the center of an object’s circular path is called centripetal force. Newton’s second law can be applied to find the magnitude of this force: Fc = mac. The equation for centripetal acceleration can be combined with Newton’s second law to obtain the following equation: Fc = mvt2/r

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Centripetal Force

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Centripetal Force Centripetal force is simply the name given to the net force on an object in uniform circular motion. Any type of force or combination of forces can provide this net force. Example: Friction between a race car’s tires and a circular track is a centripetal force that keeps the car in a circular path. Example: Gravitational force is a centripetal force that keeps the moon in its orbit.

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Centripetal Force Because centripetal force acts at right angles to an object’s circular motion, the force changes the direction of the object’s velocity. If this force vanishes, the object stops moving in a circular path and instead, moves along a straight line path that is tangent to the circle.

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Rotating Systems Think about the feelings you experience as you made a sharp turn in your vehicle. If you make a sharp turn to your right, you are thrown against the door to the left. If centripetal force is always toward the center of the circular path, why wouldn’t you be thrown toward the inside of the car rather than the outside? A popular explanation is that a force must push you outward. Many times this force is called “centrifugal force,” but to lessen confusion, we will refrain from using this term.

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Rotating Systems So let’s explain the “false force” that is centrifugal force. Before you begin to make your turn, your body is following a straight-line path. As the car enters the turn, your inertia makes to tend to move along the original straight-line path. This movement is in accordance with Newton’s first law, which states that the natural tendency of a body is to continue moving in a straight line.

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Rotating Systems If a large centripetal force acts on you, you will move in the same direction as the car. The origin of this force is the force of friction between you and the car seat. Think about it: if your seat was slippery, and the door wasn’t there, you’d slide right out! This gives you a great reason not to Armor-All your seats in a Jeep with no doors!

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**Inertia, not Centrifugal Force!**

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**Simulated Gravity Using Centripetal Force!**

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