A solid, horizontal cylinder of mass 11.0 kg and radius 1.70 m rotates with an angular speed of 5.10 rad/s about a fixed vertical axis through its center. A 1.21 kg piece of putty is dropped vertically onto the cylinder at a point 0.600 m from the center of rotation and sticks to the cylinder. Determine the final angular speed of the system.
The law of conservation of angular momentum
I₁•ω₁ = I₂•ω₂
I₁ =mR²/2, ω₁ = 5.1 rad/s
I₂ = mR²/2 +m₀•R². ω₂ = ?
(mR²/2) • ω₁=(mR²/2 +m₀•R²)•ω₂
Solve for “ ω₂ “
The value I am getting is saying its too large, how large is your value?
The law of conservation of angular momentum
I₁•ω₁ = I₂•ω₂
I₁ =mR²/2, ω₁ = 5.1 rad/s
I₂ = mR²/2 +m₀•r². ω₂ = ?
(mR²/2) • ω₁=(mR²/2 +m₀•r²)•ω₂
ω₂ =(mR²/2) • ω₁/(mR²/2 +m₀•r²)=
=m•R²•ω₁/(m•R²+2 •m₀•r²)=
=11•1.7²•5.1/(11•1.7² +2•1.21•0.6²) =
=4.96 rad/s
To determine the final angular speed of the system, we can apply the principle of conservation of angular momentum.
1. The initial angular momentum of the system is given by:
L_initial = I_cylinder * w_cylinder
where I_cylinder is the moment of inertia of the cylinder and w_cylinder is its initial angular speed.
2. The moment of inertia of a solid cylinder rotating about its central axis is given by:
I_cylinder = (1/2) * m_cylinder * r^2
where m_cylinder is the mass of the cylinder and r is its radius.
3. Plugging in the given values, we can calculate the initial angular momentum:
L_initial = (1/2) * (11.0 kg) * (1.70 m)^2 * 5.10 rad/s
4. When the piece of putty sticks to the cylinder, the moment of inertia of the system changes. The new moment of inertia can be calculated by adding the moment of inertia of the cylinder and the putty separately:
I_new = I_cylinder + I_putty
5. The moment of inertia of a point mass rotating about an axis at a distance r from the mass is given by:
I_point mass = m * r^2
where m is the mass of the point mass and r is its distance from the axis.
6. Plugging in the given values, we can calculate the moment of inertia of the putty when it sticks to the cylinder:
I_putty = (1.21 kg) * (0.600 m)^2
7. Finally, the final angular momentum of the system is given by:
L_final = I_new * w_final
where w_final is the final angular speed of the system.
8. Since angular momentum is conserved, we can equate the initial and final angular momenta:
L_initial = L_final
9. Solving for w_final, we get:
w_final = L_initial / I_new
10. Plug in the values for L_initial and I_new, and calculate w_final to find the final angular speed of the system.
Note: Make sure the units are consistent throughout the calculations, i.e., mass in kilograms, radius in meters, and angular speed in radians per second.