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.
Apply the vector equation
dw/dt = a
where w is the angular velocity and a is the angular acceleration.
a has components -21.55i (which points west) and 13.11 j (which points north).
w starts out at 140 i. After 4.6 seconds
w (t=4) = 40.87 i + 60.31 j
The magnitude does not agree with your answer. The direction tangent can be obtained from the ratio of the components.
Magnitude = 72.9
To determine the direction of the angular velocity west of north in degrees, we need to find the angle formed by the vector representing the angular velocity and the north direction.
Let's break down the steps:
1. Find the initial angular velocity vector: Given that the initial angular velocity is pointing east, we can represent it as a vector along the positive x-axis with a magnitude of 140 rad/s. The vector is given by v_i = 140i, where i is the unit vector in the x-axis direction.
2. Find the angular acceleration vector: Given that the angular acceleration is pointing 68 degrees west of north, we can represent it as a vector with a magnitude of 35 rad/s^2. To convert the direction to a vector, we need to consider the angle with the positive y-axis. Since the direction is west of north, the angle with the positive y-axis is 270 + 68 = 338 degrees. We can represent the angular acceleration vector as a unit vector u_a in the direction of 338 degrees and scale it by the magnitude of 35 rad/s^2: a = 35u_a.
3. Find the final angular velocity vector: Use the formula for angular velocity with constant angular acceleration:
v_f = v_i + a*t, where t is the time interval of 4.6 seconds.
Since v_i = 140i and a = 35u_a, plugging in the values we have:
v_f = 140i + (35u_a)*4.6
4. Calculate the magnitude and direction: To determine the magnitude of the final angular velocity, we can take the magnitude of the vector v_f using the Pythagorean theorem:
|v_f| = sqrt((140)^2 + (35)^2*(4.6)^2)
This gives us the magnitude of the angular velocity as approximately 215.94 rad/s.
To find the direction, we need to determine the angle made by the vector v_f with the positive y-axis. We can use inverse tangent to find this angle:
angle = arctan((35*(4.6))/(140))
This gives us an angle of approximately 10.95 degrees. Since the direction is west of north, we need to subtract this angle from 90 degrees. Therefore, the direction of the angular velocity is approximately 79.05 degrees west of north.
To determine the direction of the angular velocity after the angular acceleration, we can use vector addition. We are given that the initial angular velocity points east and has a magnitude of 140 rad/s. The angular acceleration points 68 degrees west of north and has a magnitude of 35 rad/s^2.
First, convert the angular acceleration from degrees to radians:
Angular acceleration (α) = 68 degrees * π/180 ≈ 1.19 rad
Next, we will consider the angular velocity as a vector quantity. The initial angular velocity can be represented as:
ω1 = 140 rad/s * (1 unit, east)
The angular acceleration can be represented as:
α = 35 rad/s^2 * (1 unit, 68 degrees west of north)
To find the final angular velocity, we can add these vectors using vector addition. However, since the direction is given in degrees, we need to convert it to radians.
Angular velocity (ω2) = ω1 + α
= 140 rad/s * (1 unit, east) + 35 rad/s^2 * (1 unit, 68 degrees west of north)
Converting the direction to radians:
68 degrees * π/180 ≈ 1.19 rad
Now we can calculate the final angular velocity:
ω2 = 140 rad/s * (1 unit, east) + 35 rad/s^2 * (1 unit, 1.19 rad west of north)
To determine the direction of the angular velocity relative to north, we can use the inverse tangent function:
Direction = atan(ω2 west/ ω2 north)
Substituting the values:
Direction = atan(1.19/ 1)
Using a calculator, we find that the direction is approximately 48 degrees west of north.
Therefore, the magnitude of the angular velocity is 61 rad/s, and its direction is approximately 48 degrees west of north.