A 26-g rifle bullet traveling 220 m/s embeds itself in a 3.8-kg pendulum hanging on a 2.7-m-long string, which makes the pendulum swing upward in an arc, Determine the vertical and horizontal component of the pendulum's maximum displacement

displacements are 0.776m, 0.114m

Explanation:

We were given mass of 26-g rifle bullet , then we can convert to Kg since

Momentum is conserved here.

The initial momentum before impact = (Mi * Vi)

Where Mi= initial given mass

Vi=initial velocity given

= 0.026 * 220 = 5.72 kgm/s

The final momentum after impact is (Mf * Vf )

Mf= final mass

5.72=( 3.82* Vf )

= 5.72/ 3.82

= 1.497 m/s

the speed of the pendulum bob with bullet afterwards= 1.497 m/s

the total energy after the collision is the addition of the kinetic energy of the bob+bullet and the potential energy of the bob and bullet, potential energy can be taken as zero.

M = 3.82 kg the mass of the bob containing the bullet

E(total) = ¹/₂MV² = 1/2 * (3.82kg)*(1.497m/s)² = 4.280J

When the Bob got to highest point the kinetic energy is zero and the potential energy is due to the increase in height of the bob, and the addition of the potential and kinetic energies still equal the total energy from before

E(total) = Mgh + 0 = Mgh = 4.280J

solving for h and substituting,

h = 4.280 J/(9.8m/s^2*3.82kg) = 0.114 m

Since the height is found,we the angle of the pendulum at the top of the swing can also be determined

A = arccos[(2.7 - 0.114) / 2.7] or A = 16.71degrees

Since A is known, the displacement along the horizontal axis can be calculated as

x = 2.7* sin(A) = 0.776m

therefore, displacement is 0.776m, 0.114m

the vertical and horizontal component of the pendulum's maximum displacement are displacement is 0.776m, 0.114m