Determine the Velocity and Acceleration of the Rotating Rod

In summary, the body in question is a slender rod rotating about a fixed axis through point O with the given angular properties of ω = 4.6 rad/s and α = 4.4 rad/s2. Using the equations v = wXr and a = αXr + wX(wXr), the instantaneous velocity and acceleration of point A were determined to be V_A = 1.24i + 2.869j and a_A = -12.01i + 8.449j, respectively. A mistake was initially made by using the radius of the circular part of the rod instead of the diameter when calculating the distance from the straight part of the rod to point A.
  • #1
Northbysouth
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2

Homework Statement


The body is formed of slender rod and rotates about a fixed axis through point O with the indicated angular properties. If ω = 4.6 rad/s and α = 4.4 rad/s2, determine the instantaneous velocity and acceleration of point A.

I have attached an image of the question

Homework Equations



v = wXr

a = αXr + wX(wXr)

The Attempt at a Solution



r = <0.5sin(24)i + 0.23cos(24)i +0.5cos(24)j - 0.23sin(24)j>

r = <0.4135i + 0.3632j + 0k>

w = 4.6k

va = <0i + 0j + 4.6k> X <0.4135i + 0.3632j + 0k>

VA = 1.6707i + 1.9021j

But it says this is wrong. What am I missing?

I know for the second part that I need to use:

aA = αXr + wX(wXr) but I can't seem to get wXr right.

Any advice would be greatly appreciated.
 

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  • #2
well I see that you are using 0.23m as the distance from the straight part of the rod to the point A. However, I think 0.23m is the radius of the cirular part of the rod, so then the distance you should be using would be the diameter 0.46m.
 
  • #3
You are correct. Thank you. My final correct answers are:

V_A = 1.24i + 2.869j

a_A = -12.01i + 8.449j
 

1. How do you determine the velocity of a rotating rod?

The velocity of a rotating rod can be determined by dividing the distance traveled by the time it took to travel that distance. This is known as the average velocity. However, if you want to know the instantaneous velocity at a specific point on the rod, you can use calculus to find the derivative of the position function at that point.

2. What is angular velocity?

Angular velocity is a measure of how fast an object is rotating around a fixed point or axis. It is typically measured in radians per second (rad/s) and is calculated by dividing the change in angle by the change in time.

3. How is acceleration related to velocity in a rotating rod?

Acceleration in a rotating rod is caused by a change in either the speed or direction of the velocity. The direction of the acceleration vector is always perpendicular to the velocity vector, pointing towards the center of rotation. The magnitude of the acceleration is equal to the square of the velocity divided by the radius of rotation.

4. What factors affect the velocity and acceleration of a rotating rod?

The velocity and acceleration of a rotating rod can be affected by several factors, including the length and mass of the rod, the distance from the axis of rotation, and the applied torque or force. Friction and air resistance can also affect the velocity and acceleration of a rotating rod.

5. How can the velocity and acceleration of a rotating rod be used in real-world applications?

The velocity and acceleration of a rotating rod can be used in various real-world applications, such as analyzing the motion of a spinning top or gyroscope, determining the speed and trajectory of a rotating object in space, and designing mechanical systems that involve rotational motion, such as engines and turbines.

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