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At the instant shown, determine the angular velocity and angular acceleration of the gyroscope wheel. Also, determine the velocity and acceleration of point

We define the following:

At the instant shown, the unit vectors

Let's find the angular acceleration of the gyroscope wheel.

To do this, it is easiest to find the angular acceleration of each angular velocity vector (and then add them together). You do this by examining how each of the angular velocity vectors (

First, calculate the angular acceleration of the angular velocity vector

where

Second, calculate the angular acceleration of the angular velocity vector

Third, calculate the angular acceleration of the angular velocity vector

(there is no change in direction of the angular velocity vector

The angular velocity of the gyroscope wheel is:

The angular acceleration of the gyroscope wheel is:

Note: The above expressions for

Note that we are following the right-hand rule here. If any of the vectors

There are some instances where the kinematics of the problem is unknown (not prescribed as it is here), so you cannot directly calculate the angular acceleration the way it is done here. Instead you have to treat it as an unknown in the problem. For an example of this, see the physics of bowling.

Let's now determine the velocity and acceleration of point

The general velocity equation for point

Where:

Since point

Let's first determine the velocity of point

Where:

Since point

and

Therefore, we have

Due to the kinematics of the gyro top, the angular velocity of the rod is not affected by the angular velocity of the gyroscope wheel. Therefore, the angular velocity of the rod is determined by setting

The position vector from point

Substitute the above two equations into equation (3) to get the velocity

From before, the general velocity equation for calculating the velocity of point

The position vector from point

We can now calculate the velocity of point

The general acceleration equation for point

Where:

Since point

Let's first determine the acceleration of point

Where:

Since point

and

Therefore, we have

From before, the angular velocity of the rod is

Due to the kinematics of the gyro top, the angular acceleration of the rod is not affected by the angular acceleration of the gyroscope wheel. Therefore, the angular acceleration of the rod is given by setting set

From before, the position vector from point

Substitute the above three equations into equation (5) to get the acceleration

From before, the general acceleration equation for calculating the acceleration of point

where

From before, the position vector from point

We can now calculate the acceleration of point

Note that equations (3), (4), (5), (6) use vector cross product multiplication. To evaluate the vector cross product for two general vectors

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