Circular Motion: Understanding Newton's Concepts of Gravity and Motion in Space, Slides of Physics

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Circular Motion

A Cannon on a Mountain

• Back to Galileo one more time… imagine a powerful cannon shooting horizontally from a high mountaintop:
• The path falls 5 m below a horizontal line in one second.

After Traveling 8 Kilometers in 1 second…

• The cannonball’s velocity has slightly changed direction, adding about g = 10 m/sec downwards, so the angle of change is given by tanθ =10/8000.
• BUT the Earth’s surface underneath the cannonball has turned by precisely the same amount—and so has the direction of gravity!
• The cannonball finds itself in exactly the same situation it began in : moving parallel to the surface, perpendicular to gravity, at the same height.
• So what happens next? docsity.com

Newton’s Own Picture

• Newton realized that at the right initial speed, above the atmosphere, the cannonball would circle indefinitely, accelerating towards the Earth constantly, but staying at the same height. (^) Link to animation

Dynamics of Circular Motion

• Constant speed circular motion has acceleration of constant magnitude but always changing direction: it points at all times to the center of the circle.
• So from , to maintain steady circular motion, a body must experience a net force of constant magnitude directed always to the center of the circle.

F = ma

Low Earth Orbit

• Newton had discovered the path of a satellite in low Earth orbit!
• For a circular orbit close to Earth’s surface, is just.
• So the speed for low orbit motion is : that’s 8 km/sec, round the Earth in 80 minutes.
• Newton’s next question: why does the Moon circle the Earth? Could it be the same reason? The force of gravity extends to the Moon?

F = ma

mg = mv^2 / r

v^2 = rg

Basic Moon Facts

• The apple accelerates downwards 3,600 times faster than the Moon.
• The Moon is 60 times further from the center of the Earth than the apple is.
• What did Newton conclude from those facts? http://dallasvintageshop.com/?p=

The Inverse Square Law of Gravity

• If the force of gravity has decreased by a factor of 3,600 on increasing the distance from the center of the Earth by a factor of 60, Newton concluded that the Earth’s gravitational force
• This is the inverse square law of gravity.
• We’ll get back to gravity in the next lecture …

2

F

r

But why mess with toys—just do it!

http://www.youtube.com/watch?v=wiZoVAZGgsw&NR=

Is this for real?

Clicker Question

If the loop track has a radius of 6 meters, approximately how fast must the car be going at the top to stay on the track?

A. About 8 m/s (18 mph)

B. About 12 m/s

C. About 16 m/s

D. About 24 m/s

Clicker Question Answer

If the loop track has a radius of 6 meters, approximately how fast must the car be going at the top to stay on the track?

A. About 8 m/s (18 mph) v^2 = rg = 60

B. About 12 m/s

C. About 16 m/s

D. About 24 m/s

Clicker Question

• If the driver has mass m , and the speed is just high enough to stay in contact with the track coasting, what is the normal force the seat exerts on him as the car enters the bottom of the loop?

A. mg

B. 2 mg

C. 5 mg

D. 6 mg

Centripetal and Centrifugal…

Circular motion is maintained by a force directed to the center of the circle: this is called the centripetal force. But if the frame of reference is itself rotating (and hence an accelerating, noninertial frame) Newton’s Laws are different: in that frame, there is an apparent force tugging outwards from the center—the centrifugal force. (Note: We’ll avoid that frame!) (^) http://imgs.xkcd.com/comics/centrifugal_force.pngdocsity.com