Confused abt derivation of centripetal acceleration

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SUMMARY

The discussion centers on the concept of centripetal acceleration and the relationship between the arc length (s) and the chord length (c) in the context of isosceles triangles. Participants clarify that as the angle θ approaches zero, the arc length s approaches the chord length c, despite the arc being longer. This is established through geometric relationships and limits, specifically using the properties of triangles and trigonometric identities. The conclusion emphasizes that for small angles, the arc length and chord length converge.

PREREQUISITES
  • Understanding of basic trigonometry, particularly sine and cosine functions.
  • Familiarity with the properties of isosceles triangles.
  • Knowledge of limits in calculus, specifically as angles approach zero.
  • Concept of tangents in relation to circles.
NEXT STEPS
  • Study the derivation of centripetal acceleration in circular motion.
  • Explore the relationship between arc length and chord length in geometry.
  • Learn about the application of limits in calculus, focusing on small angle approximations.
  • Investigate the properties of isosceles triangles and their applications in physics.
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Students of physics, mathematics enthusiasts, and educators looking to deepen their understanding of centripetal acceleration and geometric relationships in circular motion.

toforfiltum
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A comparison of corresponding parts of these two similar isosceles triangles yields

cc7ee24b-30e0-4cc9-80e2-47c7f4d1086b.gif

cce1bb03-f244-4e0f-a7d9-240c4cc0e010.gif

velocity triangle
displacement triangle

4562c164-1631-4934-9267-96a3a33872d1.gif


where

|-vo| = |vf| = v
In a small time interval Δt, the arc length s → c.


Okay, so I got this from Physics LAB, and I don't understand the last statement that states s tending to c. How could s even tend to c when the tangential velocity at A is not even directed towards c?
 
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toforfiltum said:
How could s even tend to c when the tangential velocity at A is not even directed towards c?

Do you understand what s→c means ?
 
Qwertywerty said:
Do you understand what s→c means ?
Not really, just that s is the arc and c is the cord. So I think that means the length of s gets very close to that of c.
 
toforfiltum said:
So I think that means the length of s gets very close to that of c.

Yes , it means that the length of the arc and the chord almost reach the same value ( obviously with arc greater than chord , but still getting close to the same value ) .
 
Qwertywerty said:
Yes , it means that the length of the arc and the chord almost reach the same value ( obviously with arc greater than chord , but still getting close to the same value ) .
Well, I don't get why the length of the arc tends to that of the chord.
 
toforfiltum said:
Well, I don't get why the length of the arc tends to that of the chord.

It doesn't , we just take such a value of arc and chord .
I'm sorry , do you know what a tangent is ?
 
Qwertywerty said:
It doesn't we just take such a value of arc and chord .
I'm sorry , do you know what a tangent is ?
A straight line that is perpendicular to one point in a circle.
 
Well, calculate the surface of triangle and compate it to the arc,you have an isocele triangle, the height of this triangle is h = r*cos(θ/2),It's surface is S = r2sin(θ)/2, now take this as θ→0 c = S/h = r2sin(θ)/(2*r*cos(θ/2)), use the fact that sin(θ) = 2*cos(θ/2)*sin(θ/2), c = r2sin(θ)/(2*r*(sin(θ)/2sin(θ/2)) = 2*r*sin(θ/2) = rθ = s, thus as θ → 0, c→s
 

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