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I know that terminal velocity equals to some multiple of radius square. What is the physics behind this relationship. Is it just because as terminal velocity increases, the surface area increases and thus, a larger part of the object comes into contact with the fluid?

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Is it just because as terminal velocity increases, the surface area increases?

I would say that you have the cause and effect backwards. The terminal velocity does not change the radius; the radius changes the terminal velocity. As the object size (radius) is increased, the weight increases (and the net force causing it to fall = weight - buoyancy force). To counter the increased net force, you need a higher friction force, and higher friction force is obtained by a higher terminal velocity.

thus, a larger part of the object comes into contact with the fluid?

The terminal velocity is related to objects that are completely submerged, so the portion of the object in contact with the fluid is the same.

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Yes. Remember that terminal velocity comes from equilibrium between gravity and viscous friction, which depends on the size of the object.

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Terminal velocity arises when net force on the body is zero. Thus Buoyant force + viscous force = mg We have the buoyant force as Vdg if d is density of fluid and V is volume of the body . According to Stoke's law the viscous force as $ 6π \eta rv $ Thus both buoyant force and viscous force both depends on the size

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Yes a larger radius means a larger surface area, which means a stronger drag force. The net forces are gravity pulling the object down, and drag pushing the object up. When the net force is 0, as in the force from gravity is equal to the force of drag, the object is traveling at its terminal velocity.

As radius increases, the force of drag increases.

terminal velocity (not considering buoyancy) is equal to the root of ((2 * m * g) / (density * area * drag coefficient))

As area, fluid density, or the drag coefficient increase, the terminal velocity decreases.

As mass or gravity increases, the terminal velocity increases.

Or, terminal velocity increases when the mass or the acceleration due to gravity increase, or when the density, area or drag coefficient decrease.

A falling object doesn't change its size, shape or mass easily. gravity and fluid density change with altitude though, and falling objects will slow down as the atmospheric density increases and the gravity decreases during the descent. Skydivers have some control over there shape, and can speed up or slow down during their descent by changing shape and therefore the drag coefficient.

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