Question

A bug flying horizontally at 0.85 m/s collides and sticks to the end of a uniform stick hanging vertically. After the impact, the stick swings out to a maximum angle of 5° from the vertical before rotating back. If the mass of the stick is 10 times that of the bug, calculate the length of the stick.

Answer 1

Given = 0.85 m/s

=5°  

Calculate the angular momentum of the system in the states A and B. Note that in state A , the only thing moving is the bug and therefore , the moment of inertia is that of a particle . Once the Bug is attached to the rod , the moment of inertia of the system is the one of the rod plus the moment of inertia of the particle ( the bug ) .

= = m L

= 13 m /3

Since the collision between the bug and the rod is inelastic, then the angular momentum is conserved from A to B.To substitute and into the equation for the conservation of angular momentum, and solve for the angular velocity express it as a function of the velocity and the length of the rod L.

=

m L = 13 m

=3 /13 L

Obtain the center of mass of the system in state B , measured from the lower extreme of the rod.

= 5L/11

Obtain the mechanical energy of the system in states B and C ( and respectively). Note that we have set the gravitational potential energy reference to be zero at the height of the center of mass in state B, and so there's no potential energy in state B. Since there are no non-conservative forces acting on the system between B and C, then the mechanical energy is conserved.

= 13 m /6

= 6 m g L (1-cos ​​​​​​)

=

from above equations

= 36 g (1-cos ​​​​​​) /13L

(3 /13 L) ^2 = 36 g (1- cos ​​​​​​)

The length of the Rod L

L = / 52 g ( 1- cos ​​​​​​)

Substituting given Values

L = (0.85)^2 /52 * 9.8 (1- cos 5)

=0.3757 m

The length of the stick is 0.3757m