A mechanics problem with moving and fixed coordinate systems

[mech-eng]

1. A point P is described in terms of a fixed coordinate system XYZ with unit vectors I,J,K and a moving coordinate system xyx with unit vectors i,j,k.at a given instant the location of the origin
of the moving system is 80I-90J.the velocity of P relative to moving system is 50i+45j;the radius of point P is 3i+4j in the moving system, which rotates at angular velocity w=100k. Find the velocity of point P in the fixed system if the x-axis is rotated 30 counterclockwise from the X axis.

Homework Equations

R=R₀+r
R=R_0xI+R_0yJ+R_0zk+r_xi+r_yj+r_zk
R^=R_^0xI+R^_0yJ+R^_0zK+r^_xi+r^_yj+r^_zk+r_xi^+r_yj^+r_zk^, here ^ represents derivative,i.e it is dot.

The Attempt at a Solution

v=wxr , x:cross product symbol v=300j-400i I don't know where I should use this v and its relation with another V which is 50i +45 j as given in the problem.In this problem I don't clearly understand whether w is belongs to radius or moving coordinates.

 

[KataKoniK]

Hello,

I can help clarify some of the concepts in this problem. Firstly, the given problem is asking for the velocity of point P in the fixed coordinate system, which means we need to find the velocity of point P in terms of the fixed unit vectors I, J, and K.

To do this, we can use the formula v = wxr, where v is the velocity, w is the angular velocity, and r is the radius vector. In this case, the angular velocity w is given as 100k, which means it is only in the k direction. This indicates that the rotation is only happening around the z-axis.

Next, we need to find the radius vector r in terms of the fixed coordinate system. This can be done by using the formula r = R - R0, where R is the position vector of point P in the moving coordinate system and R0 is the position vector of the origin of the moving system in the fixed system.

Using the given information, we can find R0 = 80I - 90J and R = 3i + 4j. Substituting these values into the formula, we get r = (3i + 4j) - (80I - 90J) = -80I + 90J + 3i + 4j.

Now, we can substitute the values of w and r into the formula v = wxr to get the velocity vector in terms of the fixed coordinate system. This gives us v = (100k) x (-80I + 90J + 3i + 4j) = 300j - 400i.

So, the velocity of point P in the fixed coordinate system is 300j - 400i. This is the same as the velocity given in the moving system, which is 50i + 45j, but with a 30 degree counterclockwise rotation.

I hope this helps clarify the problem. Let me know if you have any further questions. Good luck with your studies!