Find the center of mass with respect to 0, of uniform semicircular wire of mass M and Radius R (as shown in figure)
Find the center of mass with respect to 0, of uniform semicircular wire of mass M and Radius R (as shown in figure)
The correct option is B (0,2Rπ)
Take its centre as the origin. The COM must be in the plane of the wire i.e., x-y plane.
Now how do we choose the small element of the wire?
First the element should be so defined that we vary the element to cover the whole wire.
Secondly, if we are interested in∫xdm, the x-coordinates of different part of the element should only infinitesimally differ in range.
Selectingthe Element:
Take a radius making an angle θ with the x-axis and rotate it further by an angle dθ. This gives an element of length Mg.
As you can see,
If θ = 0, element is situated nearthe right edge of the wire and as θ is gradually increased to π,
the element takes all positions on the wire i.e., the whole wire is covered. The coordinates of the elements are (R cos θ, R sin θ)
(0,2Rπ)
Take its centre as the origin. The COM must be in the plane of the wire i.e., x-y plane.
Now how do we choose the small element of the wire?
First the element should be so defined that we vary the element to cover the whole wire.
Secondly, if we are interested in∫xdm, the x-coordinates of different part of the element should only infinitesimally differ in range.
Selectingthe Element:
Take a radius making an angle θ with the x-axis and rotate it further by an angle dθ. This gives an element of length Mg.
As you can see,
If θ = 0, element is situated nearthe right edge of the wire and as θ is gradually increased to π,
the element takes all positions on the wire i.e., the whole wire is covered. The coordinates of the elements are (R cos θ, R sin θ)
