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1. The problem statement, all variables and given/known data

Derive the magnitude and direction of the linear acceleration at the tip of the stick, ignore the effect of gravity.

2. Relevant equations

r = length of stick

a = dv/dt

3. The attempt at a solution

I first find the x and y components of v, then I took the derivative to find ax and ay:

vx = rω cos(θ)

vy = rω sin(θ)

ax = rω ( -sin(θ) dθ/dt ) = rω ( -sin(θ) (-ω) ) = rω^{2}sin(θ)

ay = rω ( cos(θ) dθ/dt ) = rω ( cos(θ) (-ω) ) = -rω^{2}cos(θ)

I recognized that ω is -dθ/dt since it is in the opposite direction. This shows that the magnitude of a is rω^{2}. If I were to draw the vector on the graph, it would also show that a is pointing toward the hinge.

But how would I show that a is always pointing toward the hinge?

Is this argument valid:

Let q be the a vector from the tip to the center, then

qx = sin(θ)

qy = -cos(θ)

Now, a • q = rω^{2}cos(ф) = rω^{2}, where ф is the angle between the two vectors and ф=0 in this case. Therefore a and q are in the same direction.

Is there a simpler way to show this?

- Thanks

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# Homework Help: Derive the magnitude and direction of the linear acceleration

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