A wheel spins on a horizontal axis, with an angular spped of 140rad/s and with its angular velocity pointing east. Determine the magnitude of the angular velocity after an angular acceleration of 35 rad/s^2, pointing 68 degress west of north, with a time of 4.6 seconds. I found that the magnitude of the angular velocity is 61 rad/s, but how do i determine the direction of its angular velocity west of north in degrees.
I answered this already
I do not understand what you mean by
"The direction tangent can be obtained from the ratio of the components."
arctan (West component)/(North component) = angle west from north
The west component is the negative of the east component
To determine the direction of the angular velocity west of north in degrees, you can use vector addition.
First, convert the given direction of 68 degrees west of north to a standard coordinate system. A positive rotation in a standard coordinate system is counterclockwise, so a rotation of 90 degrees represents north, and a rotation of -90 degrees represents south.
Since the direction is west of north, it will be -90 degrees plus the given angle of 68 degrees.
So, the direction of the angular acceleration is -90 + 68 = -22 degrees.
Next, consider the magnitude of the angular acceleration, which is given as 35 rad/s^2.
Since the angular acceleration is pointing in a specific direction, you need to add this vector to the initial vector of angular velocity.
To do this, you can use vector addition by treating the vectors as components in the horizontal and vertical directions.
Given the magnitude of the initial angular velocity as 140 rad/s, you can write it as:
Initial angular velocity (horizontal component) = 140 cos(0) = 140
Initial angular velocity (vertical component) = 140 sin(0) = 0
Similarly, you can represent the angular acceleration as:
Angular acceleration (horizontal component) = 35 cos(-22)
Angular acceleration (vertical component) = 35 sin(-22)
After finding the components, you can use vector addition to find the new magnitude and direction of the angular velocity.
Magnitude of the final angular velocity = sqrt[(Initial angular velocity (horizontal component) + Angular acceleration (horizontal component))^2 + (Initial angular velocity (vertical component) + Angular acceleration (vertical component))^2]
Direction of the final angular velocity = tan^-1[(Initial angular velocity (vertical component) + Angular acceleration (vertical component))/(Initial angular velocity (horizontal component) + Angular acceleration (horizontal component))]
By substituting the known values into the formulas, you can calculate the final magnitude and direction of the angular velocity.