The angular velocity of a rigid body rotating about an axis of rotation is
w = 5^i + 2^j - 4^k rad. per. sec. Determine the linear velocity of point B
on the body with a position vector r = 3^i - ^j + 3^k ft. directed from a
point on the axis of rotation.
Would the solution not simply be V= w X r
where w=5i+2j-4k
and r=3i-1j+3k
Giving (5i+2j-4k) X (3i-1j+3k) => Linear Velocity =2i-27j-11k ft per sec
The solution seems a little too simple.
Your solution is indeed correct. The formula for calculating the linear velocity of a point on a rigid body is the cross product between the angular velocity vector (w) and the position vector (r) of the point.
In this case, you have correctly identified that w = 5i + 2j - 4k rad/s and r = 3i - j + 3k ft. To calculate the cross product, you need to determine the resulting vector by calculating the determinants of the i, j, and k unit vectors along with the corresponding components of w and r.
Using the formula, you would get:
V = (5i + 2j - 4k) x (3i - j + 3k)
= (5 * (2 * 3 - (-4 * 3)) - 2 * (5 * 3 - (-4 * 3)) + (-4) * (5 * (-1) - (2 * 3))) * i
+ (5 * (3 * 3 - (-4 * 3)) - 3 * (5 * 3 - (-4 * 3)) + (-4) * (5 * (-1) - (2 * 3))) * j
+ (5 * (3 * (-1) - (2 * 3)) - (-4) * (5 * 3 - (-4 * 3)) + (-4) * (5 * (-1) - (2 * 3))) * k
Simplifying this expression, you get:
V = 2i - 27j - 11k ft/s
So your solution, V = 2i - 27j - 11k ft/s, is correct.