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Circular Motion Like Projectile Motion, Circular Motion is when objects move in two directions at the same time.

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**Think of a ball on a string.**

Twirl the object around your head. As the object moves, Newton’s 2nd Laws says that the object wants to continue moving in a straight line (inertia) with constant velocity. However, the string exerts an inward force on the ball. As long as there is a tension in the string, the ball will continue to circle around your hand. If you release the string, the ball will fly off in a straight line tangent to the string (the radius). It is the ball’s inertia that makes it continue in a straight line. The inward force on the ball acting through the string is called the centripetal force.

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**Centripetal (inward) Force**

These are the two properties which govern circular motion: Inertia Centripetal (inward) Force Inertia wants to keep it moving outward, perpendicular to the string. Centripetal Force keeps pulling it inward. Together they make it move in a circular direction.

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**What about Centrifugal Force?**

There is NO SUCH THING ! Myth BUSTED ! Here’s where the Myth began: When you twirl your ball on the string, you feel the ball pulling on your hand. People believed this was an outward force on the ball. But you know enough to understand this better. By Newton’s Third Law of Motion: Action Force: hand pulls on object Reaction Force: object pulls on hand So yes, the ball does pull your hand outward. This is the reaction force of you pulling inward on the ball. These equal but opposite forces keep the ball the same distance from your hand as it revolves around it. However, the only force acting on the ball is the inward force caused by the tension in the string. (you pulling inward)

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So, if there is NO Centrifugal Force pushing you outward, why do you get pushed outward on the Music Express ride at the Amusement Park? Answer: Technically, you don’t get pushed outward. As you go around, you have inertia and your body wants to continue moving in a straight line. However, the Music Express ride has a Centripetal Force acting on it pulling the car inward. So basically, the Music Express ride car is pushing INWARD on you. (NOT you pushing outward on the Music Express ride car.) Where do you feel the push - to your inside or outside? Think about it !

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**So let’s go back and consider the ball on the end of the string.**

Assume you are spinning it around your head at constant velocity. Even though the speed may not be changing, the ball’s direction is always changing. Since velocity and acceleration are vectors, we say that the ball’s velocity and acceleration are constantly changing. Knowing Newton’s 2nd Law of Motion, F=m*a And knowing that the centripetal force is directed inward, we know that the ball’s acceleration is also directed inward. Because of this, we call it centripetal acceleration or radial acceleration (because it is directed inward along the radius) So, when an object experiences circular motion, it has acceleration even if it is travelling at constant speed. It has acceleration because it is always changing direction. Its acceleration is directed toward the center of the circular path.

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**Let’s take a look at that ball on the end of the string:**

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**And NEVER forget Newton’s 2nd Law of Motion: F = m * a**

So: if you wanted to find the Centripetal Force on our ball on a string; 𝑭=𝒎 𝒗 𝟐 𝒓 For example: a ball (4kg) is attached to the end of a 2 m string and is whirled in a horizontal circle with centripetal acceleration of 15m/s2. What is the speed of the ball? What is the centripetal force keeping the ball in a circular path?

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**Let’s consider a car making a circular turn on a level (unbanked) road:**

The inward force pulling the car around the turn results from the friction between the tires and the road. This friction force points toward the center of the turn, and is responsible for turning the car. This is the Centripetal Force. The car also has inertia – the desire to not turn but continue going straight. Remember, there is NO outward-pointing force! The friction force points inward - toward the center of the turn (circle). Yes, you feel "thrown outward," but this is due to your inertia. There is no centrifugal force!

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**The Centripetal Force is due to the friction of the car with the road. **

Example 1: A 1200 kg car is going around a curve with radius 30 meters. If the coefficient of friction between the car's tires and the road is 0.5, what is the maximum speed at which the car can make the turn? Soultion: The Centripetal Force is due to the friction of the car with the road. We know: μ = 0.5, and FN ≈ 12,000N so: Ff = 6,000 N (this is the Centripetal Force) And using our equations for Circular Motion, we find that if the car exceeds m/s it will continue sliding forward and NOT make the turn. Try the calculations yourself now . . . If v > m/s If v <= m/s

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**What happens on wet or icy roads?**

Does μ increase or decrease? Let’s assume it is just rainy and μ = 0.35 How fast can your can go and still make the turn safely? What about on an icy road when μ = 0.2 ? How fast can your car go and still make the turn safely ?

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**Let’s talk about 2 circular motion terms you need to know:**

Period: the time it takes for an object to make one full revolution. Symbol: T Unit: second Frequency: the number of revolutions in a unit of time (in this case, one second) Symbol: f Unit: hertz (hz) Mathematically: Period = 1 / frequency and frequency = 1 / Period So: T = 1 / f and f = 1 / T

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**Now let’s put together some ideas that you already know:**

You already know that 𝑣= ∆ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∆ 𝑡𝑖𝑚𝑒 and if you make a full circle, you travel 2𝜋𝑟 radians And if the time to make a full revolution is called the Period (T), THEN: 𝑣= 2 𝜋 𝑟 𝑇 This is an equation that relates velocity to both the radius of the string AND the time it takes to make one full revolution.

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**Think about this equation: v = 2πr / T**

Two kids are riding – Emily on an inside horse and Alex on an outside horse. They both make one full revolution in the same time period (T). Who was travelling faster? Emily on the inside horse or Alex on the outside horse?

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**Newton’s Universal Law of Gravitation**

Yes, it IS that BIG ! One of Newton’s GREATEST discoveries involves the force of attraction between two bodies that are close to each other. He said that there is a force of attraction between two objects and the magnitude of that force depends on how close or far apart they are. He said that as the objects get closer, the force of attraction gets larger. He also said that if the masses of the objects get larger, the force between them also gets larger.

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Here’s an example: What is the force of gravity acting on a 2,000kg satellite orbiting the Earth two earth radii from the Earth’s center? mearth= 6x1024 kg and rearth = 6380km

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**𝐹=𝐺 𝑚 𝑦𝑜𝑢 𝑚 𝑝𝑙𝑎𝑛𝑒𝑡 𝑟 2 where r = the radius of the planet**

Gravity on the surface of a planet, etc. In the case where you are on or near the surface of an extremely large mass, use Newton’s Law of Universal Gravitation. 𝐹=𝐺 𝑚 𝑦𝑜𝑢 𝑚 𝑝𝑙𝑎𝑛𝑒𝑡 𝑟 where r = the radius of the planet Since the Force on you is just your weight, and ↓F = myouag you can plug this into the equation to get: 𝑎𝑔=𝐺 𝑚 𝑝𝑙𝑎𝑛𝑒𝑡 𝑟 2 This gives you the value for acceleration due to gravity on the planet, moon, etc. Remember: G is just a constant value, 6.67 x Nm2/kg2

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**Objects in Orbit – rotational freefall**

Objects which revolve around other celestial bodies are no different from the ball on the end of the string. They are subject to Newton’s Universal Law of Gravitation. Their Centripetal Force is caused by gravity. They are subject to the Circular Motion Equations as well.

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