Physics education research, electronic materials, and the musings of Christopher Moore, Ph.D.
Physics education research, electronic materials, and the musings of Christopher Moore, Ph.D.
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The motion of various shaped objects that bob in a pool of water can be modeled by a second order differential equation derived from F = ma.
The forces acting on the object are:
1)force due to gravity,
2)a frictional force by water,
3)and a buoyant force based on Archimedes' principle: An object that is completely or partially submerged in a fluid is acted on by an upward force equals to the weight of the water it displaces.
Derive the ODE by:
1)the dependent variable is the depth z of the object's lowest point in water,take z to be negative downward so that z= -1 means 1 ft of the object has submerged.
2)Let V(z) be the submerged volume of the object
3)m be the mass of the object
4)p be the density of the water
5)g be the acceleration due to gravity
6)w be the coef of friction for water(friction is proportional to the vertical velocity of the object)
My proposed solution is :
F= -wv + pVg - mg (negative sign for friction force since it opposes the velocity, pVg is positive since it points upward, mg is downward hence it is negative)
Does my argument make sense?
THX VERY MUCH
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That seems right.
Remember the net force on an object is always:
Where .
Last edited by Astro (2010-05-17 13:04:33)
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Your expression of the net force is right. But, as I understand you were to derive a differential equation.
The problem is one dimensional. So, we can put F = m*(d^2z/dt^2) and v = dz/dt
thus the equation is
d^2z/dt^2 = -(w/m)*dz/dt + p*g*V(z)/m - g
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