Uniform Circular Motion: some help with the math proof?

In summary, The equations ##v_x = -|v|sin(θ)## and ##v_y = |v|cos(θ)## are based on the angle between the vector v and the x-axis, which is 90+θ. Using the trigonometric identities cos(A+B) = cosAcosB - sinAsinB and sin(A+B) = sinAcosB + cosAsinB, we can see that cos(90+θ) = 0 - sin(θ) and sin(90+θ) = cos(θ) + 0, which simplifies to -|v|sinθ and |v|cosθ, respectively. This proves the validity of the equations.
  • #1
babaliaris
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Screenshot-from-2019-03-22-19-16-48.png


I can not understand why ##v_x = -|v|sin(θ)## and ##v_y = |v|cos(θ)##
I'm asking about the θ angle. If i move the vector v with my mind to the origin
i get that the angle between x'x and the vector in anti clock wise, it's 90+θ not just θ. So why is he using just θ? Does the minus in v_x somehow make the result the same when using just θ?

Also I thought that sin goes with the v_y and cos goes with v_x not the opposite...
 

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  • #2
babaliaris said:
Also I thought that sin goes with the v_y and cos goes with v_x not the opposite...
If you are using the angle between v and the x-axis, that is right. So vx = |v|cos(90+θ) and vy = |v|sin(90+θ).
Do you see how that turns into -|v|sinθ and |v|cosθ?
 
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  • #3
mjc123 said:
If you are using the angle between v and the x-axis, that is right. So vx = |v|cos(90+θ) and vy = |v|sin(90+θ).
Do you see how that turns into -|v|sinθ and |v|cosθ?

No I don't, this is what I want to understand. Can you show me mathematically how is this true? I need to see it written in math (this is important) else I can't understand why.
 
  • #4
Do you know the formulae cos(A+B) = cosAcosB - sinAsinB
and sin(A+B) = sinAcosB + cosAsinB ?
 
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  • #5
mjc123 said:
Do you know the formulae cos(A+B) = cosAcosB - sinAsinB
and sin(A+B) = sinAcosB + cosAsinB ?

Oh, I forgot about that... Now i can see why.

$$
cos(90+θ) = cos(90)cos(θ) - sin(90)sin(θ) = 0 - sin(θ) \\
sin(90+θ) = sin(90)cos(θ) + cos(90)sin(θ) = cos(θ) + 0
$$

Thank you! I was truly struggling finding this one!
 
  • #6
Here is my proof:
IMG-20190322-204937.jpg
 

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Related to Uniform Circular Motion: some help with the math proof?

1. What is uniform circular motion?

Uniform circular motion is the motion of an object in a circular path at a constant speed. This means that the object is moving at the same speed and direction throughout its motion.

2. How is uniform circular motion different from non-uniform circular motion?

In uniform circular motion, the speed of the object remains constant, while in non-uniform circular motion, the speed changes at different points along the circular path. This means that the acceleration in uniform circular motion is always directed towards the center of the circle, while in non-uniform circular motion, the acceleration can be in any direction.

3. What is the formula for calculating the velocity of an object in uniform circular motion?

The formula for calculating the velocity of an object in uniform circular motion is v = (2πr)/T, where v is the velocity, r is the radius of the circular path, and T is the time it takes for the object to complete one full revolution.

4. How is centripetal force related to uniform circular motion?

Centripetal force is the force that keeps an object moving in a circular path. In uniform circular motion, the centripetal force is equal to the mass of the object multiplied by the square of its velocity divided by the radius of the circular path, or Fc = mv²/r.

5. Can you provide an example of an object in uniform circular motion?

An example of an object in uniform circular motion is a satellite orbiting around the Earth. The satellite is constantly moving at a constant speed in a circular path around the Earth, with the Earth's gravitational force acting as the centripetal force to keep it in orbit.

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