How to find the acceleration with polar coordinates?

In summary, the conversation discusses finding the velocity and acceleration of a carriage moving on a horizontal guide due to the rotation of a drum and the shortening of a connecting cable. The equations and attempts at a solution are provided, including the use of polar coordinates and vectors. The solution involves differentiating the equation for velocity and using the relation between r and θ to find the acceleration.
  • #1
TonyTheTech
1
0

Homework Statement



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The quality of the image is bad so here's the statement:

For an interval of motion the drum of radius b turns clockwise at a constant rate ω in radians per second and causes the carriage P to move to the right as the unwound length of the connecting cable is shortened. Use polar coordinates r and θ and serive expressions for the velocity v and acceleration a of P in the horizontal guide in terms of the angle θ. Check your solution with time of the relation x2+h2=r2

Homework Equations



I first found that the velocity of the carriage is v=bω/sin(θ)

The Attempt at a Solution



I attempted to directly derivate the equation which give me -bωcos(θ)/sin2(θ)

However, in the book answers, the answer is supposer to be b2ω2/h *cot3(θ).

I think I have to do something with the vectors, like derivate v=vrer + vθeθ but I don't understand much.
 
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  • #2
It seems to me that you just forgot a term in the chain rule to differentiate your expressior for the speed. Realize that as r is changing, so is [itex]\theta[/itex], so [itex]\theta[/itex] is also a function of time.

You have [itex] v = -\dfrac{b\omega}{\sin{\theta}} [/itex]

[itex]a=\dfrac{dv}{dt} = \dfrac{b\omega\cos{\theta}}{\sin{\theta}^2}\dfrac{d\theta}{dt}[/itex]

There is probably a simpler way to do this, but to obtain [itex]\frac{d\theta}{dt}[/itex] you can realize that [itex] r*\cos{\theta} = h[/itex], where h is a constant. If you differentiate both sides, you get

[itex] \dfrac{dr}{dt}*cos{\theta} + r(-\sin{\theta})\dfrac{d\theta}{dt} = 0 [/itex]

You can solve for [itex]\dfrac{d\theta}{dt}[/itex] and plug it into the expression for a, it gives you the right answer :) Realize that [itex]\frac{dr}{dt} = -b\omega[/itex], since it is simply the rate at which the rope is being pulled.

Did I make any sense? :P
 
Last edited:

Related to How to find the acceleration with polar coordinates?

1. How do you define acceleration in polar coordinates?

In polar coordinates, acceleration is defined as the rate of change of the velocity vector in the polar coordinate system. It is a measure of how quickly the magnitude and direction of the velocity of an object changes over time.

2. What is the formula for finding acceleration in polar coordinates?

The formula for calculating acceleration in polar coordinates is a = (ar, aθ), where ar is the radial acceleration and aθ is the tangential acceleration. This can also be written as a = (r̈ - rθ̇^2, rθ̈ + 2ṙθ̇), where r̈ is the second derivative of the radial position, rθ̇ is the first derivative of the tangential position, and rθ̈ is the second derivative of the tangential position.

3. How do you find the radial and tangential components of acceleration in polar coordinates?

To find the radial and tangential components of acceleration, you can use the formulas ar = r̈ - rθ̇^2 and aθ = rθ̈ + 2ṙθ̇. These components can also be calculated using the velocity components, vr and vθ, and the position components, r and θ, as ar = (vr^2 + r^2θ̇^2)^(1/2) and aθ = (rθ̇^2 + 2ṙvr + r^2vθ) / r.

4. How is acceleration represented graphically in polar coordinates?

In polar coordinates, acceleration can be represented graphically as a vector with both a magnitude and a direction. The magnitude of the acceleration vector is given by the length of the arrow, while the direction of the acceleration is given by the angle of the arrow with respect to the radial direction.

5. What are some real-world applications of finding acceleration in polar coordinates?

Finding acceleration in polar coordinates is useful in many applications, such as analyzing the motion of objects moving in a circular path, calculating the forces acting on a satellite in orbit, or studying the motion of planets in our solar system. It is also used in fields such as robotics, where polar coordinates are commonly used to control the movement of robotic arms or other mechanisms.

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