Exploring Radial Acceleration and Displacement

In summary, the conversation discussed the equations for centripetal acceleration and angular velocity, as well as the integration of the tangential velocity to find the arc length. The units of arc length are meters, while the units of ##\int \omega dt## are meters.
  • #1
jisbon
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30
Homework Statement
An object is undergoing circular motion in horizontal plane at fixed radius## r = 0.12m##
Radial acceleration is ##2+2t ##m/s
Calculate arc length the object swept through the first 2 seconds.
Relevant Equations
-
From what I understand,

##a_{r} = v_{tan}^2 /r##
##a_{r} = (r\omega)^2 /r##
##a_{r} = r\omega^2##
##\omega^2 = \frac{a_{r}}{r}##
##\omega^2 = \frac{2+2t}{0.12}##
##\omega = \sqrt{\frac{2+2t}{0.12}}##
##s =\int_{0}^{2} \sqrt{\frac{2+2t}{0.12}}##
After integrating, I still can't seem to get the correct answer which is 1.37m
Are my concepts wrong or..?
Thanks
 
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  • #2
jisbon said:
##s =\int_{0}^{2} \sqrt{\frac{2+2t}{0.12}}##
What are the units of arc length? What are the units of ##\int \omega dt##?
 
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  • #3
A
tnich said:
What are the units of arc length? What are the units of ##\int \omega dt##?
Arc length has no units, ##\omega## has a SI units : rad s−1
I realized that when I integrate ##\omega## I will get angular displacement instead of arc length. So after I get the angular displacement I can just multiply it by radius to get the arc length?

EDIT: Solved~! Thanks for the reminder
 
Last edited:
  • #4
jisbon said:
So after I get the angular displacement I can just multiply it by radius to get the arc length?
Or simply go with integrating the tangential velocity from your first expression and never care about angular velocity.
 
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  • #5
Orodruin said:
Or simply go with integrating the tangential velocity from your first expression and never care about angular velocity.
Oh yep, that's another alternative :)
 
  • #6
jisbon said:
Arc length has no units
It has dimension length, so the SI unit is metres.
 
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1. What is radial acceleration and how is it different from linear acceleration?

Radial acceleration refers to the acceleration of an object moving in a circular or curved path. It is different from linear acceleration, which refers to the change in velocity of an object moving in a straight line. Radial acceleration is always perpendicular to the velocity of the object, while linear acceleration can be in any direction.

2. How is radial acceleration calculated?

Radial acceleration can be calculated using the formula a = v^2/r, where a is the radial acceleration, v is the velocity of the object, and r is the radius of the circular path. This formula can also be written as a = ω^2r, where ω (omega) is the angular velocity of the object.

3. What is the relationship between radial acceleration and displacement?

The relationship between radial acceleration and displacement depends on the type of motion. In uniform circular motion, the radial acceleration is constant and the displacement is directly proportional to the square of the time. In non-uniform circular motion, the radial acceleration and displacement are constantly changing and can be calculated using calculus.

4. How does radial acceleration affect centripetal force?

Radial acceleration is directly related to centripetal force, which is the force that keeps an object moving in a circular path. The greater the radial acceleration, the greater the centripetal force needed to maintain the circular motion. Without sufficient centripetal force, an object will move in a straight line tangential to the circle.

5. What are some real-world examples of radial acceleration and displacement?

Some real-world examples of radial acceleration and displacement include the motion of planets orbiting around the sun, the motion of a car on a circular track, and the motion of a rollercoaster. Other examples include the motion of a satellite in orbit around Earth and the motion of a ball being swung on a string. In all of these examples, the object experiences both radial acceleration and displacement as it moves in a circular or curved path.

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