An 30 g rifle bullet traveling at 390 m/s buries itself in a 4 kg pendulum hanging on a 2.8 m long string, which makes the pendulum swing upward in an arc. Determine the maximum vertical displacement of the pendulum (height).(g

Explanation:

Mass of bullet 30g=30/1000=0.03kg

Speed of bullet =390m/s

Mass of pendulum is 4kg

Total mass of bullet and pendulum, =4.03kg

Length of pendulum is L= 2.8m

Then, momentum is conserved

Momentum before collision = momentum after Collision

Mass of bullet ×Velocity of bullet=combine mass of bullet and pendulum × velocity of pendulum

0.03×390=4.03×V

V=0.03×390/4.03

V=2.9m/s

Then the Kinetic energy of pendulum is given as

K.E=½mv²

K.E=½×4.03×2.9²

K.E=16.95J

conservation of energy states that energy can neither be created nor destroy but can be transferred from one form to another.

Let acceleration due to gravity be

g=9.81m/s²

The vertical displacement of the pendulum (h)=?

Potential energy of the pendulum is given as

P.E=mgh

P.E=4.03×9.81×h

P.E=39.53h

So the P.E of the pendulum equals K.E of the pendulum

P.E=K.E

39.53h=16.95

h=16.95/39.53

h=0.43m

Knowing the height we can find the angle of the pendulum at the top of the swing:

θ=arcos(l-h)/l

θ=arcos(2.8-0.43)/2.8

θ=arcos(0.8464)

θ=32.2°

From A we get the displacement along the horizontal axis:

The maximum vertical displacement is

x=h Sinθ

x=0.43 Sin32.2

x=0.23m

So the displacement are (0.23,0.43)m

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