Solution Found!
Center of Percussion In the previous problem, suppose a
Chapter 10, Problem 108GP(choose chapter or problem)
In the previous problem, suppose a small metal ball of mass \(m=2M\) is attached to the rod a distance d from the pivot. The rod and ball are released from rest in the horizontal position. (a) Show that when the rod reaches the vertical position, the speed of its tip is
\(v_\mathrm{t}=\sqrt {3gL} \sqrt{\frac{1+4(d/L)}{1+6(d/L)^2}}\)
(b) At what finite value of d/L is the speed of the rod the same as it is for \(d=0\)? (This value of \(d/L\) is the center of percussion, or “sweet spot,” of the rod.)
Equation Transcription:
Text Transcription:
m=2M
vt=sqrt{3gL}sqrt{frac{1+4(d/L)}{1+6(d/L)^2}}
d/L
d=0
d/L
Questions & Answers
QUESTION:
In the previous problem, suppose a small metal ball of mass \(m=2M\) is attached to the rod a distance d from the pivot. The rod and ball are released from rest in the horizontal position. (a) Show that when the rod reaches the vertical position, the speed of its tip is
\(v_\mathrm{t}=\sqrt {3gL} \sqrt{\frac{1+4(d/L)}{1+6(d/L)^2}}\)
(b) At what finite value of d/L is the speed of the rod the same as it is for \(d=0\)? (This value of \(d/L\) is the center of percussion, or “sweet spot,” of the rod.)
Equation Transcription:
Text Transcription:
m=2M
vt=sqrt{3gL}sqrt{frac{1+4(d/L)}{1+6(d/L)^2}}
d/L
d=0
d/L
ANSWER:
a.)
Step 1 of 3
We have to show that when the rod reaches the vertical position, the speed of its tip is given by
The speed of the rod at its tip is given by the expression,
where,
angular speed at the tip in rad/s
The angular speed can be found by making use of the conservation of mechanical energy for the system.
(constant)
where,
refers to the horizontal position
refers to the vertical position
Here,
and
are the masses of the rod
and ball respectively.
is the length of the rod
is Moment of Inertia of the rod
and ball combination =
is the distance at which the ball is attached to the
rod from pivot