An object of mass 10 moves with position function r(t)=<1+3cos(t),4cos(t),−1+5sin(t)> Find the force vector acting on this object at time t. Show that it points from the object towards the point (1,0,−1).

Tyson Atkins 2022-10-14 Answered
An object of mass 10 moves with position function r ( t ) =< 1 + 3 cos ( t ) , 4 cos ( t ) , 1 + 5 sin ( t ) > Find the force vector acting on this object at time t. Show that it points from the object towards the point ( 1 , 0 , 1 )
I believe that I have found the correct force vector: F ( t ) = 10 < 3 cos t , 4 cos t , 5 sin t >
However, I am not sure how to figure out how to show that it points from the object towards the point ( 1 , 0 , 1 )
Any help would be greatly appreciated.
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Answers (1)

na1p1a2pafr
Answered 2022-10-15 Author has 16 answers
What you need to do is describe the line along F at time t. You know that it is going through r ( t ). So the equation of this line is
l α = r ( t ) + α F ( t ) ,   α R
Plugging in what you calculated so far,
l α ( t ) =< 1 + 3 cos t 30 α cos t , 4 cos t 40 α cos t , 1 + 5 sin t 50 α sin t >
Now let's calculate the intersection by saying that l α ( t 1 ) = l β ( t 2 ). Then
1 + 3 ( 1 10 α ) cos t 1 = 1 + 3 ( 1 10 β ) cos t 2 4 ( 1 10 α ) cos t 1 = 4 ( 1 10 β ) cos t 2 1 + 5 ( 1 10 α ) sin t 1 = 1 + 5 ( 1 10 β ) sin t 2
If we want this intersection to be time independent, we see that α = β = 0.1 Plugging in this value of α, you get a single point < 1 , 0 , 1 >
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