Motion in Polar Coordinates problem

In summary, the conversation revolves around a problem in polar coordinates where a particle is described by the curve r=be^[Zcot(a)] and the task is to show that its velocity and acceleration vectors have angles a and 2a with OP (O is the origin). Z is the angle the position vector makes with the x axis, and its derivative with respect to time is a constant, w. B and a are also constants. The conversation includes a suggested method for solving the problem by simplifying the expression for velocity and using the dot product to find the angle between the vectors. Eventually, the problem is solved with the help of the suggestion.
  • #1
Master J
226
0
Been looking over past exam questions and came across this one. Its in polar coordinates:

A particle P describes the curve r=be^[Zcot(a)].

Show that the velocity and acceleration vectors have angles a and 2a with OP (O is the origin).

Z is actually theta, the angle the position vector makes with the x axis. Its derivative to time is w, a constant here. B and a are constants.


I'm stumped!


I work out v as: bcot(a)we^(Zcot(a)).[e_1] + bwe^('').[e_2]

Even with them worked out I am stumped as to show that^^^^

Any pointers?

Cheers guys!
 
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  • #2
Hi Master J,

Master J said:
Been looking over past exam questions and came across this one. Its in polar coordinates:

A particle P describes the curve r=be^[Zcot(a)].

Show that the velocity and acceleration vectors have angles a and 2a with OP (O is the origin).

Z is actually theta, the angle the position vector makes with the x axis. Its derivative to time is w, a constant here. B and a are constants.


I'm stumped!


I work out v as: bcot(a)we^(Zcot(a)).[e_1] + bwe^('').[e_2]

I would suggest first simplifying this expression for v, using the expression for r given in the problem. Then, since they ask for the angle between two vectors, that suggests using the dot product (since one definition of the dot product explicitly includes the angle between the vectors). Do you get the answer?
 
  • #3
SOlved! Thanks for the help.
 
  • #4
Sure, glad to help!
 

What is motion in polar coordinates?

Motion in polar coordinates is a mathematical concept used to describe the movement of an object in a two-dimensional plane. It involves measuring an object's distance from a fixed point (known as the pole) and its angle of rotation (known as the polar angle).

How do you convert polar coordinates to Cartesian coordinates?

To convert from polar coordinates (r, θ) to Cartesian coordinates (x, y), you can use the following equations:
x = r * cos(θ)
y = r * sin(θ). These equations use the trigonometric functions cosine and sine to determine the x and y coordinates based on the given polar angle and distance from the pole.

What is the difference between linear motion and circular motion in polar coordinates?

In linear motion, an object travels in a straight line, while in circular motion, an object follows a curved path. In polar coordinates, linear motion is represented by a constant polar angle and a changing distance from the pole, while circular motion is represented by a constant distance from the pole and a changing polar angle.

How do you calculate the velocity and acceleration in polar coordinates?

To calculate the velocity in polar coordinates, you can use the equation v = r * ω, where r is the distance from the pole and ω is the angular velocity. To calculate the acceleration, you can use the equation a = (r * α) + (r * ω²), where α is the angular acceleration.

What are some real-world applications of motion in polar coordinates?

Motion in polar coordinates is used in various fields such as physics, engineering, and astronomy. It is commonly used to describe the movement of objects in circular or elliptical orbits, the motion of pendulums, and the rotation of objects around a fixed point. It is also used in navigation systems, robotics, and computer graphics.

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