The 10 -g bullet having a velocity of 800 m / s is fired into the edge of the 5-kg disk as show

The 10 -g bullet having a velocity of 800 m / s is fired...

The 10 -g bullet having a velocity of $800 \mathrm{~m} / \mathrm{s}$ is fired into the edge of the $5-\mathrm{kg}$ disk as shown. Determine the angular velocity of the disk just after the bullet becomes embedded into its edge. Also, calculate the angle $\theta$ the disk will swing when it stops. The disk is originally at rest. The $\operatorname{rod} A B$ has a mass of $3 \mathrm{~kg}$.

Key Concepts

Energy Conservation in Rotational Motion

This principle involves the transformation of rotational kinetic energy into potential energy and vice versa. When a rotating system such as a disk swings upward, its kinetic energy is converted into gravitational potential energy, allowing for the determination of the maximum swing angle when the system momentarily comes to rest.

Conservation of Angular Momentum

This principle states that in the absence of external torques, the total angular momentum of a system remains constant. In collision problems, even if kinetic energy is lost, the angular momentum before and after the collision must be equal, which makes it essential for determining post-collision rotation rates.

Inelastic Collision

In an inelastic collision, the colliding bodies stick together after impact, resulting in a loss of kinetic energy. However, the conservation of angular momentum still applies, allowing for the calculation of subsequent rotational motion despite the energy loss.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on the mass distribution relative to the axis of rotation. It is crucial for relating the angular velocity of a system to its rotational kinetic energy and angular momentum.

Transcript

00:01 In this question, the mass moment of inertia of the desk igd equals half times 5 times 0 .4 square equals 0 .4 kilogram meter square and igr equals of 1 over 12 times 3 times 2 power 2 equals 1 kilo gram meter square also vp2 equals omega 2 5 .92 multiply omega 2 as we said and vd2 equals omega 2 multiply 2 .4 finally we are equal to omega 2 multiply 1 so now sigma sigma h a at 1 equals sigma h a at 2 so we have mb times vb 1 times r b1 equals i g d omega 2 plus m d vd 2 r d plus r d plus r d igr omega 2 plus mr vr 2 multiply r plus mb b b2 times b b times r b2 if we substitute by the numbers we will get 0 .01 times 800 times 2 equals 0 .4 omega 2 plus 5 omega 2 multiply 2 square plus 1 times omega 2 plus 3 multiply omega 2 multiply root 5 .92 square.

03:15 From this equation we can calculate omega 2 which is equal 0 .5773 gradient per second since the system is required to stop finally so this is omega 2 so t 3 should be equal to 0 and t2 equal to to half multiply igd omega 2 square plus half md vt2 square plus half igr omega 2 square plus half mr multiply vb2 square if we substitute we will have half multiply 0 .4 multiply 0 .573 or power 2 plus half multiply 5 5 multiply 0 .573 multiply 2 .2 plus half multiply 1 .5773 multiply 2 .2 plus half multiply 1 .51773 multiply 2 plus half x .573 power 2 plus half multiply 3 multiply 0 .573 plus half multiply 0 .01 multiply 0 .733 square root 5 .92 square so t2 in this case will be equal 5 .549 jules so datum is set as indicated so phi equals 10 inverse .4 over 2 .4 which is equal 9 .4623 degrees hinds y d equals 2 .4 or cosine theta and yr equals cosine theta and yb equals root 5 .92 cosine theta minus 9 .4623 thus the gravitational potential energy of the desk rod and pull it with with reference to the datum.

06:53 So vg d equals md multiply g multiply yd so we have 5 multiply 9 .81 1 .81 multiply 0 .81 multiply negative 2 .4 cosine ceta which is equal negative 117 .72 cosine ceta.

07:21 And vgr equal mrg multiply yr so we have 3 times 9 .81 multiply xeta multiply negative cata equals negative 29 .43 cosine ceta and vgb equals m b equals m bg yb equals 0 .01 times 9 .81 multiply negative root 5 .92, cosine, ceta minus 9 .4, 623...

Please give Ace some feedback