A. a particle moves around the circle x^2+y^2 = 1 in such a way that the x coord rate of change is dx/dy= y
B) IS THE PARTICLE MOVING COLCIKWISER OR COUNTERCLOCKWISE AROUND THE CIRCLE
C) FROM THE RT TRIANGLE ONE CAN SEE THAT SIN(THETA) = Y AND COS(THETA) = X
USE THESE RELATIONS TO SHOW THAT THE ANGLE THETA IS CHANGING AT A CONST. RATE AND THEN FIND THAT RATE
B. Something is fishy about this queation.
dx/dy is negative in the first quadrant and third quadrant. y is positive in the first and second quadrant. Furthermore,
dx = y dy means that
x = (y^2/2) + C
where C is any constant.
The particle is not following the circle at all. It is follwing a parabola. Nothing can be said about the direction or rate it is moving because time does not appear in your equations.
Are you sure you copied the problem correctly?
Should you have written dy/dt = x ?
agree with drwls
I started working on this, following the precise information as you stated it
x^2 + y^2 = 1
2x dx/dy + 2y dy/dy = 0
dx/dy = -y/x , but you said dx/dy = y
y = -y/x
x = -1
at this point I stopped.
To determine whether the particle is moving clockwise or counterclockwise around the circle, we need to examine the information given. In this case, the rate of change of the x-coordinate, dx/dy, is equal to y.
Let's break down the problem into steps:
A) To find out if the particle is moving clockwise or counterclockwise, we need to determine the sign of the rate of change. Since dx/dy = y, we can tell that if y is positive, dx/dy is also positive, indicating counterclockwise motion. If y is negative, then dx/dy would be negative, indicating clockwise motion.
B) From the given information, we can see that the particle is moving counterclockwise around the circle.
C) In a right triangle formed by the particle's position on the circle and the x-axis, we can establish the following relations:
sin(theta) = y
cos(theta) = x
We can differentiate both sides of these equations with respect to time (assume t is the independent variable):
d(sin(theta))/dt = d(y)/dt
d(cos(theta))/dt = d(x)/dt
Using the chain rule and the given information dx/dy = y, we can substitute:
cos(theta) * d(theta)/dt = dy/dt
-sin(theta) * d(theta)/dt = dx/dt
-y * d(theta)/dt = dx/dt
Since dx/dt = dx/dy * dy/dt = y * dy/dt, we can further simplify:
-y * d(theta)/dt = y * dy/dt
By canceling out the common factor of y from both sides:
-d(theta)/dt = dy/dt
From this equation, we can see that the rate of change of the angle theta, d(theta)/dt, is equal to the rate of change of y, dy/dt, but with a negative sign. This means that the angle theta is changing at a constant rate, but in the opposite direction.
To find the exact value of this rate, we would need additional information about the rate of change of y, dy/dt.
To determine whether the particle is moving clockwise or counterclockwise around the circle, you can analyze the given information about the rate of change.
In this case, you are provided with the rate of change of the x-coordinate with respect to the y-coordinate: dx/dy = y.
Since dx/dy > 0 when y > 0 and dx/dy < 0 when y < 0, we can conclude that as y increases, the x-coordinate also increases.
To understand this visually, you can think of it as the particle starting at the point (1, 0) on the right side of the circle. As it moves upwards (i.e., y increases), the x-coordinate also increases, pushing the particle counterclockwise around the circle.
Therefore, the particle is moving counterclockwise around the circle.
Now, let's move on to part C of your question. We are asked to use the trigonometric relations sin(theta) = y and cos(theta) = x to show that the angle theta is changing at a constant rate and find that rate.
From the given circle equation, x^2 + y^2 = 1, we can rewrite it as x = cos(theta) and y = sin(theta).
Differentiating both sides of the x = cos(theta) equation with respect to y (using the chain rule), we get:
dx/dy = d(cos(theta))/dy = -sin(theta) * d(theta)/dy
Since dx/dy = y, we can substitute y = sin(theta) into the equation:
sin(theta) * d(theta)/dy = -sin(theta)
Simplifying the equation, we find:
d(theta)/dy = -1
This shows that the angle theta is changing at a constant rate of -1.
Therefore, the rate at which the angle theta is changing is constant, and it is equal to -1.