Write Vector Expression in n-t and x-y coordinates of Acceleration

AI Thread Summary
The discussion focuses on deriving the vector expression for the acceleration of a simple pendulum's mass center in both n-t and x-y coordinates at a specific angle. The user successfully calculated the tangential acceleration (at) as 18.795 ft/sec² and the normal acceleration (an) as 20.69928 ft/sec². However, they encountered difficulties converting these components into Cartesian coordinates (i and j). After community input, the user adjusted their calculations for the x and y components, resulting in -25.589i for the x-component and -11.265j for the y-component. The thread highlights the importance of correctly applying geometry and trigonometric relationships in vector transformations.
Northbysouth
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Homework Statement


Write the vector expression for the acceleration a of the mass center G of the simple pendulum in both n-t and x-y coordinates for the instant when θ = 66° if θ'= 2.22 rad/sec and θ"= 4.475 rad/sec2

I have attached an image of the question.

Homework Equations


an = v2/r = rθ2 = vθ'

at = v' = rθ'


The Attempt at a Solution



I've managed to calculate en and et correctly

at = (4.2 ft)(4.475 rad/sec2) = 18.795 ft/sec2

at = 18.795ft/sec2

For an I calculated velocity first:

v = rθ' = (4.2ft)(2.22 rad/sec)
v = 9.324 ft/sec

Hence an = (9.324 ft/sec)(2.22 rad/sec)
an = 20.69928 ft/sec2

Unfortunately, I'm now having difficulty with finding the velocity in terms of i and j.

I had thought that I could use geometry to do it:

atcos(90-66) = -17.77 i
atsin(90-66) = 7.644 j

But the system says it's wrong. Help is greatly appreciated.
 

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Why don't you try to express en and et in terms of ex and ey ? Doing that, you can find the total acceleration, a = an + an , in terms of its projections on x and y axis.* e i is the unit vector in the direction of the subscript "i".
 
Northbysouth said:

Homework Statement


Write the vector expression for the acceleration a of the mass center G of the simple pendulum in both n-t and x-y coordinates for the instant when θ = 66° if θ'= 2.22 rad/sec and θ"= 4.475 rad/sec2

I have attached an image of the question.

Homework Equations


an = v2/r = rθ2 = vθ'

at = v' = rθ'

The Attempt at a Solution



I've managed to calculate en and et correctly

at = (4.2 ft)(4.475 rad/sec2) = 18.795 ft/sec2

at = 18.795ft/sec2

For an I calculated velocity first:

v = rθ' = (4.2ft)(2.22 rad/sec)
v = 9.324 ft/sec

Hence an = (9.324 ft/sec)(2.22 rad/sec)
an = 20.69928 ft/sec2

Unfortunately, I'm now having difficulty with finding the velocity in terms of i and j.

I had thought that I could use geometry to do it:

atcos(90-66) = -17.77 i
atsin(90-66) = 7.644 j

But the system says it's wrong. Help is greatly appreciated.
attachment.php?attachmentid=55292&d=1359837236.png


Both n and t components contribute to each of the i and j components.
 
@SammyS: You were right. To find the i component I did the following:

-atcos(24) -ancos(66) = -25.589i

For j:

atsin(24) - ansin(66) = -11.265 j

Thanks everyone.
 
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