Deriving Orbital Motion from Collisions in Travel along Chords

[LucidLunatic]

Homework Statement

Inscribe N chords of equal length within a circle of radius r. A body of mass m travels at constant speed v along the chords and is perfectly reflected in each collision with the circle (there is no momentum change tangent to the circle). Show that the radial momentum change per collision is Δp =mvsinθ, where θ is the angle between tangent and chord.

Homework Equations

Δp_r = p_r¹-p_r⁰

The Attempt at a Solution

We can write
p⁰ = mvsinθ \hat{r} + mvcosθ\hat{\theta}
p¹ = -mvsinθ \hat{r} + mvcosθ\hat{\theta}

therefore Δp = -2mvsinθ.

This seems like a fairly simple problem, so I'm not sure where I'm going wrong. The later steps of this problem then show that the net centripetal force obeys F = mv^2/r as N→∞, but that depends on getting the correct answer here. What am I missing? Note: this is me going through problems in a book, not an actual homework assignment.

 

[haruspex]

Homework Statement

Inscribe N chords of equal length within a circle of radius r. A body of mass m travels at constant speed v along the chords and is perfectly reflected in each collision with the circle (there is no momentum change tangent to the circle). Show that the radial momentum change per collision is Δp =mvsinθ, where θ is the angle between tangent and chord.

Homework Equations

Δp_r = p_r¹-p_r⁰

The Attempt at a Solution

We can write
p⁰ = mvsinθ \hat{r} + mvcosθ\hat{\theta}
p¹ = -mvsinθ \hat{r} + mvcosθ\hat{\theta}

therefore Δp = -2mvsinθ.

This seems like a fairly simple problem, so I'm not sure where I'm going wrong. The later steps of this problem then show that the net centripetal force obeys F = mv^2/r as N→∞, but that depends on getting the correct answer here. What am I missing? Note: this is me going through problems in a book, not an actual homework assignment.

I agree with your Δp (except the sign depends on how you care to define +ve direction), and from it I obtain F = mv²/r.

 

[rude man]

Dumb EE's observation:

The above calculatios (posts 1 and 2) were made under the assumption that the radial component of momentum is also conserved. But that is not stated.

Let M = mass of circle and V be its + radial velocity after collision. Then if the system is isolated form the environment (e.g. on a frictionless horizontal plane):

Assume MV = +3mv sinθ
the answer would be correct, since then for the radial direction

p₁ = +mv sinθ
p₂ = -mv sinθ + MV = -mv sinθ + 3mv sinθ = 2mv sinθ
so that p₂ - p₁ = 2mv sinθ - mv sinθ = mv sinθ.

In other words the problem has no solution unless the physical aspects of the circle are defined:

circle's mass
system isolated form the environment or not?
etc.

Comments?

 

[haruspex]

Dumb EE's observation:

The above calculatios (posts 1 and 2) were made under the assumption that the radial component of momentum is also conserved. But that is not stated.

Let M = mass of circle and V be its + radial velocity after collision. Then if the system is isolated form the environment (e.g. on a frictionless horizontal plane):

Assume MV = +3mv sinθ
the answer would be correct, since then for the radial direction

p₁ = +mv sinθ
p₂ = -mv sinθ + MV = -mv sinθ + 3mv sinθ = 2mv sinθ
so that p₂ - p₁ = 2mv sinθ - mv sinθ = mv sinθ.

In other words the problem has no solution unless the physical aspects of the circle are defined:

circle's mass
system isolated form the environment or not?
etc.

Comments?

I presumed the exercise related to kinetic theory of gases, so the mass of the circle is effectively infinite.

 

[rude man]

I presumed the exercise related to kinetic theory of gases, so the mass of the circle is effectively infinite.

I agree, that makes sense.