11 Angular Momentum
11 Chapter Review
Key Terms
- angular momentum
- rotational analog of linear momentum, found by taking the product of moment of inertia and angular velocity
- law of conservation of angular momentum
- angular momentum is conserved, that is, the initial angular momentum is equal to the final angular momentum when no external torque is applied to the system
- precession
- circular motion of the pole of the axis of a spinning object around another axis due to a torque
- rolling motion
- combination of rotational and translational motion with or without slipping
Key Equations
| Velocity of center of mass of rolling object | [latex]{v}_{\text{CM}}=R\omega[/latex] |
| Acceleration of center of mass of rolling object | [latex]{a}_{\text{CM}}=R\alpha[/latex] |
| Displacement of center of mass of rolling object | [latex]{d}_{\text{CM}}=R\theta[/latex] |
| Acceleration of an object rolling without slipping | [latex]{a}_{\text{CM}}=\frac{mg\,\text{sin}\,\theta }{m+({I}_{\text{CM}}\text{/}{r}^{2})}[/latex] |
| Angular momentum | [latex]\mathbf{\overset{\to }{l}}=\mathbf{\overset{\to }{r}}\times \mathbf{\overset{\to }{p}}[/latex] |
| Derivative of angular momentum equals torque | [latex]\frac{d\mathbf{\overset{\to }{l}}}{dt}=\sum \mathbf{\overset{\to }{\tau }}[/latex] |
| Angular momentum of a system of particles | [latex]\mathbf{\overset{\to }{L}}={\mathbf{\overset{\to }{l}}}_{1}+{\mathbf{\overset{\to }{l}}}_{2}+\cdots +{\mathbf{\overset{\to }{l}}}_{N}[/latex] |
| For a system of particles, derivative of angular momentum equals torque |
[latex]\frac{d\mathbf{\overset{\to }{L}}}{dt}=\sum \mathbf{\overset{\to }{\tau }}[/latex] |
| Angular momentum of a rotating rigid body | [latex]L=I\omega[/latex] |
| Conservation of angular momentum | [latex]\frac{d\mathbf{\overset{\to }{L}}}{dt}=0[/latex] |
| Conservation of angular momentum | [latex]\mathbf{\overset{\to }{L}}={\mathbf{\overset{\to }{l}}}_{1}+{\mathbf{\overset{\to }{l}}}_{2}+\cdots +{\mathbf{\overset{\to }{l}}}_{N}=\text{constant}[/latex] |
| Precessional angular velocity | [latex]{\omega }_{P}=\frac{rMg}{I\omega }[/latex] |
Summary
11.1 Rolling Motion
- In rolling motion without slipping, a static friction force is present between the rolling object and the surface. The relations vCM=Rω,aCM=Rα,anddCM=RθvCM=Rω,aCM=Rα,anddCM=Rθ all apply, such that the linear velocity, acceleration, and distance of the center of mass are the angular variables multiplied by the radius of the object.
- In rolling motion with slipping, a kinetic friction force arises between the rolling object and the surface. In this case, vCM≠Rω,aCM≠Rα,anddCM≠RθvCM≠Rω,aCM≠Rα,anddCM≠Rθ.
- Energy conservation can be used to analyze rolling motion. Energy is conserved in rolling motion without slipping. Energy is not conserved in rolling motion with slipping due to the heat generated by kinetic friction.
11.2 Angular Momentum
- The angular momentum l⃗ =r⃗ ×p⃗ l→=r→×p→ of a single particle about a designated origin is the vect