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Circular Motion
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Uniform Circular Motion
Uniform Circular Motion β Traveling with a constant speed in a circular path Even though the speed is constant, the acceleration is non-zero The acceleration responsible for uniform circular motion is called centripetal acceleration We can calculate π π by relating Ξ π£ to Ξπ‘
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π£ π
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π£ = ? π π Ξπ π 0
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Ξ π£ π£ π π£ 0 π π π£ 0 Ξπ π 0
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Centripetal Acceleration
When an object travels with uniform circular motion, the acceleration always points towards the center of the circular path with: π π = π£ 2 π
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Period, Frequency, & Angular Frequency
Period (π)β The time it takes to complete one full rotation Frequency (π)β Rotations per second (or per minute) Angular Frequency (π) β Radians per second
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Period, Frequency, & Angular Frequency
Period and frequency are related by: π= 1 π There are 2π radians in one rotation, therefore: π=2π π
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Angular Frequency Equations
If an object rotates through an angle π with a radius π, how far does the object travel? Ξs π
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Angular Frequency Equations
In general: Ξs=r π and π£=π π
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Angular Frequency Equations
Combining π π = π£ 2 π and π£=π π gives: π π =π π 2
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Centripetal Acceleration - Example
A centrifuge creates a centripetal acceleration of π π 2 . The average radius of the arm of the centrifuge is π=5 ππ. How fast does the centrifuge spin in revolutions per second?
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π π =61250 π π π£= ? π=0.05 π π= ?
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Centripetal Acceleration - Example
A car drives around a level turn. The tires have a coefficient of friction of π π =0.8, and the turn has a radius of 90 π. How fast can the car go around the turn without sliding?
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π£ = ? π
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Centripetal Acceleration - Example
A car drives around an banked turn with a radius of 90 π. The turn is designed so that a car traveling 10 π/π will be able go around the turn even when the coefficient of friction is reduced to π π =0. What angle is the turn banked at?
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π£ = ? π π
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Circular Motion with Gravity
The centripetal force is any force that holds an object in circular motion. That is, any force that points inwards towards the center of a circular path. As an object goes around a loop, the forces that make up the centripetal force change.
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Circular Motion with Gravity
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Circular Motion with Gravity
How fast does a rollercoaster need to be traveling when it goes through a vertical loop with a radius of 4.0 π ?
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π=4.0 π
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Newtonβs Law of Universal Gravity
Every particle exerts an attractive force all other particles The force is given by: πΉ πΊ = πΊ π 1 π 2 π 2 πΊ is the universal gravitational constant: πΊ= β11 π π 2 π π 2
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Newtonβs Law of Universal Gravity
πΉ πΊ = πΊ π 1 π 2 π 2 Note that π, is the distance between the centers of the two masses Therefore, when standing on the surface of the Earth: π= π
πΈ +β β π
πΈ
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Using Newtonβs Law of Universal Gravity, we can show that π=9
π πΈ = kg π
πΈ = π Using this formula, we can calculate the acceleration due to gravity on planets other than Earth.
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Universal Gravity - Example
Another planet is discovered. The new planet has a radius half the radius of the Earth, R p =0.5 π
πΈ , and one-tenth the mass of Earth π π =0.1 π πΈ . What is the acceleration due to gravity on this planet?
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π
π =0.5 π
πΈ π π =0.1 π πΈ π π =?
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Orbit Any force can supply the centripetal force required to keep an object in uniform circular motion. When a satellite obits the Earth, the centripetal force is supplied by the gravitational force. πΉ π
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Universal Gravity - Example
How fast is the satellite moving when it is placed in a circular orbit around the Earth with a radius of π? The Earth has a mass of ππ.
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π= π π πΈ = ππ πΊ= β11 π π 2 π π π£= ?
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Keplerβs Third Law of Planetary Motion
Keplerβs Third Law relates the period π to the radius of the orbit π
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Keplerβs Third Law of Planetary Motion
Keplerβs Third Law of Planetary Motion Says: π 2 π
3 =ππππ π‘.
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Keplerβs Third Law - Example
An asteroid orbits the Sun with a radius that is exactly twice the radius of the Earthβs orbit around the Sun. How long does it take this asteroid to orbit the Sun?
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Keplerβs Third Law - Example
π πΈ =1 π¦πππ π
π΄ =2 π
πΈ π π΄ = ?
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