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fallingsphere

# fallingsphere - Phys374 Spring 2006 Prof Ted Jacobson...

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Unformatted text preview: Phys374, Spring 2006, Prof. Ted Jacobson Falling sphere with air drag force We considered the drag force and argued that to a good approximation it might depend just on the radius of the sphere R , the density of the air ρ , and the speed of the sphere through the air v . Only one combination of these three quantities has the dimensions of force: F drag ∝ ρR 2 v 2 . (1) We can’t infer the dimensionless coefficient. Let’s define b as the product of this unknown coefficient times ρR 2 , so F drag = bv 2 . (2) Note that the dimensions of b are M/L . If the sphere has mass m and falls in a gravitational field with grav- itational acceleration g , then Newton’s law (taking the down direction as positive) says m dv dt = mg- bv 2 . (3) If the sphere begins at rest it will initially accelerate with acceleration g . As it speeds up the drag force increases until it balances the gravity force. This will happen asymptotically, i.e. in an infinite amount of time. Let’s work out the details....
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fallingsphere - Phys374 Spring 2006 Prof Ted Jacobson...

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