Finding the acceleration in a curvilinear motion (n-t)

In summary, the mine skip is being hauled to the surface over a curved track with a shape described by y = x2/28. The skip is pulled by a cable wound around a 38-in. drum turning at a constant speed of 96 rev/min. The total acceleration of the skip at a level of 3.5 ft below the top is calculated using the formula an = v2/ρ, where v is the angular speed of the drum and ρ is the radius of curvature. After correcting for an error in assuming x=3.5 ft instead of y=3.5 ft, the final result for ρ is 15.33 ft.
  • #1
Zang

Homework Statement


The mine skip is being hauled to the surface over the curved track by the cable wound around the 38-in. drum, which turns at the constant clockwise speed of 96 rev/min. The shape of the track is designed so that y = x2/28, where x and y are in feet. Calculate the magnitude of the total acceleration of the skip as it reaches a level of 3.5 ft below the top. Neglect the dimensions of the skip compared with those of the path. Recall that the radius of curvature is given by
upload_2017-9-17_23-22-58.png


Homework Equations


an = v2/ρ

The Attempt at a Solution


I assumed that the angular speed of the drum is the same as the speed of the mine skip which is constant, so tangential acceleration would be 0.
v = 96 rev/min = 15.92 ft/s.
ρ = 15.33 ft using the given formula.
an = 15.922/15.33 = 16.53 ft/s2.
since at = 0 => a = an = 16.53 ft/s2, but it said my answer is wrong. Is that because at is not 0 and how do I find it?
 
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  • #2
If the tangential acceleration was zero the skip would not speed up ...
 
  • #3
Orodruin said:
If the tangential acceleration was zero the skip would not speed up ...
It didn't say the skip speeds up and doesn't the speed of the skip equal to the angular speed?
 
  • #4
Zang said:
It didn't say the skip speeds up and doesn't the speed of the skip equal to the angular speed?
I missed the drum and cable and thought it was accelerating due to gravity.

Can you show your work leading up to ##\rho = 15.33'##? I do not get the same result.
 
  • #5
Orodruin said:
I missed the drum and cable and thought it was accelerating due to gravity.

Can you show your work leading up to ##\rho = 15.33'##? I do not get the same result.
y = x2/28
dy/dx = x/14
at x = 3.5 ft, dy/dx = 3.5/14 = .25
d2y/dx2 = 1/14
ρ = ((1+.252)3/2)/(1/14) = (1.0625)3/2 * 14 = 1.095 * 14 = 15.33 ft
 
  • #6
The problem statement seems to indicate that y=3.5', not x=3.5'.
 
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Likes Zang
  • #7
Orodruin said:
The problem statement seems to indicate that y=3.5', not x=3.5'.
I totally missed that. Thank you so much!
 

Related to Finding the acceleration in a curvilinear motion (n-t)

1. What is curvilinear motion?

Curvilinear motion refers to the movement of an object along a curved path, rather than a straight line. This type of motion can be seen in circular or elliptical orbits, as well as in the motion of objects on roller coasters or in car races.

2. What is the difference between linear and curvilinear motion?

Linear motion involves movement in a straight line, while curvilinear motion involves movement along a curved path. In linear motion, the velocity and acceleration are constant, while in curvilinear motion, they can vary as the object follows the curved path.

3. How do you find the acceleration in curvilinear motion?

To find the acceleration in curvilinear motion, you must first determine the tangential and normal components of acceleration. The tangential acceleration is the change in speed along the curved path, while the normal acceleration is the change in direction of the object's velocity. These components can then be combined using vector addition to find the total acceleration.

4. What is the role of the centripetal acceleration in curvilinear motion?

The centripetal acceleration is the component of acceleration that points towards the center of the curved path. It is responsible for keeping the object in its curved path and preventing it from flying off in a straight line. This is why objects in circular motion must constantly accelerate towards the center, even if their speed remains constant.

5. How does the acceleration in curvilinear motion affect an object's velocity and position?

The acceleration in curvilinear motion can alter both the velocity and position of an object. If the acceleration is in the same direction as the velocity, the object's speed will increase. If the acceleration is in the opposite direction, the object's speed will decrease. Additionally, the acceleration can also change the direction of the object's velocity, causing it to move along a different curved path.

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