Physics question?

Question

A particle rotates counterclockwise in a circle w/ a radius of 7.4 m and a constant angular speed of 13 rad/s. At t = 0 the particle has an x coordinate of 4.9 m and y is greater than 0.

At t = 1.56 s
a) What is the x cooordinate of the particle (in m)?
b) what is the x component of the particle's velocity (in m/s)?
c) What is the x component of the particle's acceleration (in m/s^2)?

Answer 1

To start with, define the position of the point not in terms of x and y, but in terms of some angle z from the positive x axis. From this angle you can determine to x and y coordinates.

Define zi as the initial angle with x coordinate 4.9 and y >0. Draw a triangle with hypotenuse 7.4 (radius of the circle) and x coordinate 4.9. Determine zi:

cos(zi) = x/r = 4.9/7.4 = 0.662162162
cos-1(0.662162162) = zi = 0.847 rad

a) The formula for z in terms of w (angular velocity) is similar to the formula for distance in terms of velocity:

z = w*t + zi = 13*1.56 + 0.847 = 21.13 rad

Now to figure out the x coordinate, use this as the angle, 7.4 for the hypotenuse and solve for x:
cos(21.13) = x/7.4
x = cos(21.13)*7.4 = -4.82 m

b) The angular velocity is constant, so the tangential velocity is too. The tangential velocity (v) is equal to the angular velocity times the radius:

v = w * r = 13 * 7.4 = 96.2 m/s

The direction of the velocity is tangent to the circle, so the angle created with the x-axis is perpendicular to the position angle (z). Just add pi/2 to z to get the angle the velocity makes:

va (velocity angle) = z + pi/2 = 21.13 + 1.57 = 22.7 rad

Now determine the x component by using the velocity as the hypotenuse:

cos(va) = x / v
x = cos(va) * v = cos(22.7) * 96.2 = -73.0 m/s

c) The direction for the particle's acceleration will be toward to center of the circle of motion. The equation for acceleration is:

a = r*w^2 = 7.4*13^2 = 1251 m/s^2

The angle the acceleration makes with the x-axis is exactly opposite the angle the position makes, so az (acceleration angle):

az = z + pi = 21.13 + 3.14 = 24.27 rad

To figure out the x component of the acceleration, use the acceleration as the hypotenuse:

cos(az) = ax / a
ax = cos(az) * a = cos(24.27)*1251 = 814 m/s^2

I hope this helps!

Answer 2

The position of the particle in circular motion can be expressed as

x(t) = r*cos(w*t + q)
y(t) = r*sin(w*t + q)

where w is the angular speed, t is time, r is the amplitude (radius of orbit), and q is the phase shift. At t = 0 you have

x(0) = r*cos(q)

which can be solved for the phase shift.

Once you have the phase shift, you have the full functional form for the x and y displacements as functions of time.

For the particle's velocity, it is the first derivative of the displacement function and is expressed as

vx(t) = -r*w*sin(w*t + q) : x component
vy(t) = r*w*cos(w*t + q) : y component

while the acceleration is the second derivative of the displacement and is given by

ax(t) = -r*w^2*cos(w*t + q) : x component
ay(t) = -r*w^2*sin(w*t + q) : y component.

Be sure in doing the calculations of the trigonometric functions to make sure you do not mix up radians and degrees. (It is easy to solve for the phase shift in degrees and then add that to the w*t which is in radians.)

Answer 3

x= - 4.8m
Vx= 4.8m/s
and forgot how to calculate acceleration for a spinning particle