Equilibrium, Torque, and Rotational Stability (4A) - Chem & Physics

1. Find $\alpha$ if $\sum\tau=12\,\text{N}\cdot\text{m}$ acts on an object with $I=3.0\,\text{kg}\cdot\text{m}^2$.

   $\alpha=4.0\,\text{rad}\cdot\text{s}^{-2}$. Angular acceleration equals net torque divided by moment of inertia, per the rotational second law.

2. State the relationship between torque and angular acceleration for rotation about a fixed axis.

   $\sum \tau=I\alpha$. This rotational analog of Newton's second law relates net torque to angular acceleration via moment of inertia.

3. Identify the torque about the hinge from a $20\,\text{N}$ force applied $0.30\,\text{m}$ away at $0^\circ$ to the door.

   $\tau=0\,\text{N}\cdot\text{m}$. At $\theta=0^\circ$, $\sin(\theta)=0$, so the lever arm is zero, producing no torque.

4. Find the net torque if $+6\,\text{N}\cdot\text{m}$ (CCW) and $9\,\text{N}\cdot\text{m}$ (CW) act on the same axis.

   $\sum\tau=-3\,\text{N}\cdot\text{m}$ (clockwise). Net torque is the algebraic sum, with clockwise negative, resulting in $6 - 9 = -3\,\text{N}\cdot\text{m}$.

5. State the moment of inertia of a uniform rod of mass $M$ and length $L$ about its center (axis perpendicular to rod).

   $I=\frac{1}{12}ML^2$. Derived from integrating mass elements along the rod's length about its midpoint.

6. Find the torque magnitude if $r=0.40\,\text{m}$, $F=10\,\text{N}$, and $\theta=90^\circ$.

   $\tau=4.0\,\text{N}\cdot\text{m}$. With $\theta=90^\circ$, $\sin(\theta)=1$, so torque equals the product of $r$ and $F$.

7. State the static friction inequality that prevents slipping for a body in equilibrium on a surface.

   $f_s\le\mu_s N$. Static friction provides the necessary force up to its maximum to maintain no relative motion in equilibrium.

8. State the moment of inertia of a point mass $m$ at distance $r$ from the rotation axis.

   $I=mr^2$. For a point mass, inertia depends on mass and squared distance from the rotation axis.

9. State the moment of inertia of a thin hoop (ring) of mass $M$ and radius $R$ about its center.

   $I=MR^2$. All mass elements of the hoop are at distance $R$ from the central axis, yielding this inertia value.

10. State the formula for torque magnitude using lever arm and force magnitude.

    $\tau=rF\sin(\theta)$. Torque magnitude arises from the cross product, incorporating the perpendicular component via $\sin(\theta)$.

11. What is the lever arm (moment arm) in the torque expression $\tau=rF\sin(\theta)$?

    $r_\perp=r\sin(\theta)$. The lever arm represents the perpendicular distance from the axis to the line of action of the force.

12. What is the condition for rotational equilibrium in terms of net torque about an axis?

    $\sum \tau=0$. Rotational equilibrium is achieved when the sum of all torques about any axis is zero, preventing angular acceleration.

13. Which force component produces torque about a pivot: parallel or perpendicular to the radius vector?

    Perpendicular component: $F_\perp=F\sin(\theta)$. Torque is produced only by the force component perpendicular to the position vector from the pivot.

14. What is the definition of moment of inertia for point masses about an axis?

    $I=\sum m_ir_i^2$. Moment of inertia quantifies rotational inertia as the sum of each mass times its squared distance from the axis.

15. Find the lever arm $r_\perp$ if $r=0.50\,\text{m}$ and the force makes $30^\circ$ with $\vec{r}$.

    $r_\perp=0.25\,\text{m}$. Lever arm is $r \sin(\theta)$, where $\theta$ is between $\vec{r}$ and $\vec{F}$, yielding $0.50 \times \sin(30^\circ)=0.25\,\text{m}$.

16. What is the rotational kinetic energy of a rigid body with moment of inertia $I$ and angular speed $\omega$?

    $K=\frac{1}{2}I\omega^2$. Rotational kinetic energy is analogous to translational, substituting $I$ for $m$ and $\omega$ for $v$.

17. Identify the direction of torque given by the right-hand rule for $\vec{\tau}=\vec{r}\times\vec{F}$.

    Along $\vec{r}\times\vec{F}$ (thumb direction of right-hand rule). The right-hand rule determines the direction of the cross product $\vec{r} \times \vec{F}$, with thumb pointing along torque.

18. State the moment of inertia of a solid disk (or solid cylinder) of mass $M$ and radius $R$ about its center.

    $I=\frac{1}{2}MR^2$. Integration over the disk's uniform mass distribution results in half the inertia of an equivalent hoop.

19. What is the parallel-axis theorem for moment of inertia?

    $I=I_{\text{cm}}+Md^2$. The theorem allows calculation of inertia about a parallel axis displaced by distance $d$ from the center of mass.

20. What sign convention is commonly used for planar torques in statics problems?

    Counterclockwise $+$, clockwise $-$. This convention assigns positive to counterclockwise and negative to clockwise for consistent torque summation in 2D problems.

21. What is the SI unit of torque?

    $\text{N}\cdot\text{m}$. Torque units derive from force times perpendicular distance, yielding newton-meters in SI.

22. What is angular momentum for a rigid body rotating about a fixed axis?

    $\vec{L}=I\vec{\omega}$. For rigid body rotation about a fixed axis, angular momentum is the product of inertia and angular velocity.

23. State the moment of inertia of a uniform rod of mass $M$ and length $L$ about one end (axis perpendicular to rod).

    $I=\frac{1}{3}ML^2$. Obtained using the parallel-axis theorem from the center-of-mass inertia, adding $M(L/2)^2$.

24. What is the relationship between net external torque and angular momentum?

    $\sum \vec{\tau}_{\text{ext}}=\frac{d\vec{L}}{dt}$. This equation is the rotational equivalent of Newton's second law, linking torque to change in angular momentum.

25. What is the condition for translational equilibrium in terms of net force?

    $\sum \vec{F}=\vec{0}$. Translational equilibrium requires the vector sum of all forces acting on an object to be zero, ensuring no net linear acceleration.

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