Mechanics - Acceleration on an elliptical path

In summary, the question asks for the maximum magnitude of acceleration for a particle traveling around an elliptical path with a constant speed of 1 m/s. Using the equations given, the radius of curvature is found to be 3, with a corresponding acceleration of 1. However, the provided answer choices do not align with this result. Further examination of the question reveals that the smallest radius of curvature is being asked for, resulting in a maximum acceleration of √2 for the second question and 3/2 for the first question. These answers do not match any of the choices given. It is unclear what the intended answer or formula for calculating the maximum acceleration is.
  • #1
Kaevan807
15
0
This is a mechanics question which has come up a number of times in end of year exams in my college, I hope this is the right forum to post it in.

Homework Statement


A particle travels around an elliptical path with a constant speed of 1 m/s. If the equation
of the ellipse is (x^2)/3 + y^2 = 1
Determine the maximum magnitude of the acceleration.

A - (√3)/2
B - 3/2
C - 1/2
D - 0
E - 1

Homework Equations


If the equation = (x^2)/(a^2) + (y^2)/(b^2) = 1
Then a = (v^2)/ρ, where ρ is the radius of curvature and is equal to (a^2)/b

The Attempt at a Solution


Using the above equations gives:
ρ = 3/1 = 3
a = 1/3 which is not an answer provided?


I'm unsure whether the answer should just be zero? As there is no tangential acceleration? Or do I have the wrong formula for calculating the centripetal acceleration?

Any help or explanation would be great :)
 
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  • #2
Kaevan807 said:
Then a = (v^2)/ρ, where ρ is the radius of curvature and is equal to (a^2)/b

Is this the maximum or minimum radius of curvature of the ellipse?
 
  • #3
This is the maximum radius of curvature.
 
  • #4
Kaevan807 said:
This is the maximum radius of curvature.

But you want to find the maximum acceleration.
 
  • #5
TSny said:
But you want to find the maximum acceleration.

Yes but isn't the maximum centripetal acceleration when the particle is traveling at the point of largest radius?
 
  • #6
Kaevan807 said:
Yes but isn't the maximum centripetal acceleration when the particle is traveling at the point of largest radius?

No...
 
  • #7
Remember:
Kaevan807 said:
... a = (v^2)/ρ, where ρ is the radius of curvature ...
 
  • #8
Had an entire post written and then realized that it's the smaller radius but furthest from centre of the ellipse.

Ok so taking that into account would the answer be 1?

Minimum radius = 1

a = (v^2)/r = 1/1 = 1?Edit
Thank you, can't believe I didn't get that >.< That's a great help.
 
  • #9
Kaevan807 said:
Ok so taking that into account would the answer be 1?

Minimum radius = 1
a = (v^2)/r = 1/1 = 1?

No, the minimum radius of curvature of the ellipse is not 1. a2/b is the curvature of the ellipse at a point on the y-axis. This happens to be the maximum radius of curvature for your ellipse because the ellipse is elongated along the x-axis. Think about where the minimum curvature will occur and see if you can guess how the formula a2/b would change to get the minimum curvature.
 
  • #10
Dang just when I thought I had it.

Well I'm guessing the equation will change to b2/a?

Although that gives me 1/√3 which when subbed into a doesn't result in an answer?


Edit:

I was going under the assumption that a and b are the two radii? So whichever is smaller I just sub directly into a=v2/r, this gives me an answer for all 3 years this question has come up
 
  • #11
Kaevan807 said:
Dang just when I thought I had it.

Well I'm guessing the equation will change to b2/a?

Although that gives me 1/√3 which when subbed into a doesn't result in an answer?

I think 1/√3 is the right answer even though it's not listed as one of the choices.

I was going under the assumption that a and b are the two radii? So whichever is smaller I just sub directly into a=v2/r, this gives me an answer for all 3 years this question has come up

I hope whoever wrote the question wasn't assuming that the smallest radius of curvature would be the smaller of a and b!
 
  • #12
Written by a mechanics lecturer in my college but unfortunately I haven't had a reply to any email I've sent him. The questions the other two years were:
1. A particle travels around an elliptical path with a constant speed of 1 m/s. If the equation
of the ellipse is 3x2 + 2y2 = 2
Determine the maximum magnitude of the acceleration.

A - (√3)/2
B - √3/√2 (Answer using the radius being the smaller of a and b)
C - 1/2
D - 0
E - 1

2. A particle travels around an elliptical path with a constant speed of 1 m/s. If the equation of the ellipse is x2/2 + y2 = 1
Determine the maximum magnitude of the acceleration

A - (√3)/2
B - 3/2
C - 1/2
D - 0
E - 1 (Answer using the radius being the smaller of a and b)


As you can see neither of these has an answer which would work using the b2/a formula
 
  • #13
For #1, a is smaller than b. So, the smallest radius of curvature is a2/b = 2/3. Max acceleration is then 3/2.

For #2, b is smaller than a. So, the smallest radius of curvature is b2/a = 1/√2. Max acceleration is √2.

As you noted, neither of these corresponds to any of the selections. Hope I'm not overlooking something.
 
  • #14
Unless he meant the magnitude of the tangential acceleration? Which would be zero each time? But he doesn't say that
 

1. What is acceleration on an elliptical path?

Acceleration on an elliptical path is the rate at which an object's velocity changes as it moves along an elliptical trajectory. It is a vector quantity, meaning it has both magnitude and direction.

2. How is acceleration on an elliptical path calculated?

The acceleration on an elliptical path can be calculated using the equation a = v^2/r, where a is the acceleration, v is the velocity, and r is the radius of the elliptical path.

3. What factors affect acceleration on an elliptical path?

The acceleration on an elliptical path is affected by the velocity of the object, the radius of the elliptical path, and the gravitational force acting on the object.

4. How does acceleration on an elliptical path differ from acceleration on a circular path?

Acceleration on an elliptical path is not constant, unlike acceleration on a circular path. It changes depending on the position of the object along the elliptical trajectory.

5. What is the significance of acceleration on an elliptical path in real-life scenarios?

Acceleration on an elliptical path is important in understanding the motion of celestial bodies such as planets and satellites, which move along elliptical orbits around larger objects. It is also relevant in studying the motion of objects in roller coasters and other amusement park rides.

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