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

In summary, the vector expression for the acceleration a of the mass center G of the simple pendulum can be written as a = -25.589i - 11.265 j ft/sec2 at the instant when θ = 66°, given that θ' = 2.22 rad/sec and θ" = 4.475 rad/sec2.
  • #1
Northbysouth
249
2

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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  • #2
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".
 
  • #3
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.
 
  • #4
@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.
 
  • #5




The vector expression for the acceleration a of the mass center G of the simple pendulum can be written as follows:

In n-t coordinates:

a = an n + at t

where an is the normal component of acceleration and at is the tangential component of acceleration.

In x-y coordinates:

a = ax i + ay j

where ax is the x-component of acceleration and ay is the y-component of acceleration.

Using the given values, we can calculate the components of acceleration in both coordinate systems:

In n-t coordinates:

an = vθ' = (4.2 ft)(2.22 rad/sec) = 9.324 ft/sec
at = v' = rθ" = (4.2 ft)(4.475 rad/sec2) = 18.795 ft/sec2

Therefore, the vector expression for acceleration in n-t coordinates is:

a = (9.324 ft/sec) n + (18.795 ft/sec2) t

In x-y coordinates:

ax = atcosθ - ansinθ = (18.795 ft/sec2)cos(66°) - (9.324 ft/sec)sin(66°) = 7.644 ft/sec2
ay = atsinθ + ancosθ = (18.795 ft/sec2)sin(66°) + (9.324 ft/sec)cos(66°) = 17.77 ft/sec2

Therefore, the vector expression for acceleration in x-y coordinates is:

a = (7.644 ft/sec2) i + (17.77 ft/sec2) j

I hope this helps! Keep up the good work in your studies of physics.
 

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

1. What is a vector expression?

A vector expression is a mathematical representation of a vector quantity, which has both magnitude and direction. It is typically written as a set of coordinates or components, such as n-t (normal and tangential) or x-y (horizontal and vertical) coordinates.

2. How is acceleration represented in n-t coordinates?

In n-t coordinates, acceleration is represented as a vector with two components: normal acceleration, which is perpendicular to the object's motion, and tangential acceleration, which is parallel to the object's motion. The vector expression for acceleration in n-t coordinates can be written as (an, at).

3. Can acceleration be expressed in both n-t and x-y coordinates?

Yes, acceleration can be expressed in both n-t and x-y coordinates. In x-y coordinates, acceleration is represented as a vector with two components: horizontal acceleration (ax) and vertical acceleration (ay).

4. How do you convert between n-t and x-y coordinates?

To convert a vector expression from n-t coordinates to x-y coordinates, you can use trigonometric functions. The horizontal component (ax) can be calculated as ax = at * cos(θ), where θ is the angle between the vector and the x-axis. Similarly, the vertical component (ay) can be calculated as ay = an * sin(θ). The reverse conversion can also be done using the same trigonometric functions.

5. How is acceleration related to other vector quantities?

Acceleration is directly related to velocity and indirectly related to displacement. It is the rate of change of velocity and can be calculated by taking the derivative of the velocity vector with respect to time. In addition, acceleration is also related to force through Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration.

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