Change in Momentum in Time for Curving Motion

claimed by Elizabeth Cooper

The momentum of an object going around a circle is always changing even if its speed is constant. This is because its directions is constantly changing, therefore its velocity is changing. The change in momentum for curving motion always points toward the center of the circle.

The Main Idea

[math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] for Curving Motion

A Mathematical Model

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

A Computational Model

Model of a space craft orbiting a planet. "https://trinket.io/embed/glowscript/6d7763cad6" The arrow in red represents the [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] of the space craft.

Components of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math]

Perpendicular Component

The perpendicular component is zero if the direction of the object's motion is not changing and it is nonzero if it is changing. The perpendicular component of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] is equal to [math]\displaystyle{ {\frac{mv^2}{R}} }[/math], where m is the mass of the object, v is the velocity of the object, and R is the radius of the kissing circle.

Parallel Component

The parallel component is zero if the speed of the object is constant and it is nonzero if the object's speed is changing.

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Simple

Example: A 3kg ball is moving at a constant speed of 12m/s in a circle of radius 5m. Find the magnitude and direction of the perpendicular and parallel

components of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math].

Solution: [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{||} }[/math] is zero because the speed of the object is not changing. To solve for [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{\perp} }[/math], we can use the equation [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{\perp} }[/math] = [math]\displaystyle{ {\frac{mv^2}{R}} }[/math].

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{\perp} }[/math] = [math]\displaystyle{ {\frac{(3kg)(12m/s)^2}{5m}} }[/math]

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{\perp} }[/math] = 86.4N

Middling

Difficult

Connectedness

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History

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References

Sherwood, Bruce A. "5.7 [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] for Curving Motion." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.