A bullet of mass = 0.13 kg is fired with a velocity of = 87.1 m/s into a solid cylinder of mass = 17.1 kg and radius = 0.29 m. The cylinder is initially at rest and is mounted on a fixed vertical axis that runs through its center of mass. The line of motion of the bullet is perpendicular to the axis and at a distance = 0.0067 m from the center. Find the angular speed of the system after the bullet strikes and adheres to the surface of the cylinder.
To find the angular speed of the system after the bullet strikes and adheres to the surface of the cylinder, we can use the principle of conservation of angular momentum.
1. Find the initial angular momentum:
The initial angular momentum of the system is given by the product of the moment of inertia and the angular velocity.
Initial angular momentum = (moment of inertia of the cylinder + moment of inertia of the bullet) × initial angular velocity
The moment of inertia of a cylinder can be given as:
I_cylinder = 0.5 × mass_cylinder × radius_cylinder^2
The moment of inertia of a point mass can be given as:
I_bullet = mass_bullet × distance_bullet^2
Initial angular momentum = (0.5 × mass_cylinder × radius_cylinder^2 + mass_bullet × distance_bullet^2) × initial angular velocity
2. Find the final angular momentum:
After the bullet strikes and adheres to the surface of the cylinder, the moment of inertia changes. The moment of inertia becomes:
I_cylinder_final = 0.5 × (mass_cylinder + mass_bullet) × radius_cylinder^2
Final angular momentum = I_cylinder_final × final angular velocity
3. Apply the conservation of angular momentum:
According to the conservation of angular momentum, the initial angular momentum is equal to the final angular momentum.
(0.5 × mass_cylinder × radius_cylinder^2 + mass_bullet × distance_bullet^2) × initial angular velocity = I_cylinder_final × final angular velocity
4. Solve for the final angular velocity:
Rearrange the conservation of angular momentum equation to solve for the final angular velocity.
final angular velocity = (0.5 × mass_cylinder × radius_cylinder^2 + mass_bullet × distance_bullet^2) × initial angular velocity / I_cylinder_final
Plug in the given values to calculate the final angular velocity.
To find the angular speed of the system after the bullet strikes and adheres to the surface of the cylinder, we can use the principle of conservation of angular momentum.
Angular momentum is given by:
Angular momentum (L) = Moment of Inertia (I) × Angular Speed (ω)
Initially, the cylinder is at rest, so its initial angular momentum is zero:
Initial angular momentum (Linitial) = 0
After the bullet strikes and adheres to the cylinder, the system has a combined moment of inertia (Itotal) given by the sum of the individual moments of inertia of the bullet and the cylinder.
The moment of inertia of the bullet (Ibullet) is given by:
Ibullet = mbullet × r^2
where mbullet is the mass of the bullet and r is the distance of the bullet from the axis of rotation.
The moment of inertia of a solid cylinder (Icylinder) is given by:
Icylinder = 0.5 × mcylinder × r^2
where mcylinder is the mass of the cylinder and r is the radius of the cylinder.
The total moment of inertia (Itotal) is the sum of the individual moment of inertia of the bullet and the cylinder:
Itotal = Ibullet + Icylinder
Since the bullet adheres to the surface of the cylinder, we can write:
Itotal × ωfinal = (mbullet + mcylinder) × ωfinal
Solving for ωfinal, the angular speed of the system after the collision:
ωfinal = (mbullet + mcylinder) / Itotal
Now, let's plug in the given values:
mbullet = 0.13 kg (mass of the bullet)
mcylinder = 17.1 kg (mass of the cylinder)
r = 0.0067 m (distance of the bullet from the center)
R = 0.29 m (radius of the cylinder)
First, we need to calculate the individual moments of inertia:
Ibullet = 0.13 kg × (0.0067 m)^2
Icylinder = 0.5 × 17.1 kg × (0.29 m)^2
Next, we calculate the total moment of inertia:
Itotal = Ibullet + Icylinder
Finally, we can calculate the angular speed:
ωfinal = (mbullet + mcylinder) / Itotal
By substituting the numerical values and solving the equation, we can find the angular speed of the system after the bullet strikes and adheres to the cylinder.