Rotational Equilibrium and Newton's First Law - AP Physics C Mechanics

1. What does a non-zero net torque cause?

   A change in angular velocity. Net torque produces angular acceleration.

2. What is the relationship between force and torque?

   Torque is the rotational effect of a force, $\tau = r \times F \times \text{sin}(\theta)$. Torque is force creating rotation about a pivot.

3. State Newton's First Law in rotational form.

   An object will not rotate or will rotate at constant angular velocity unless acted upon by a net torque. Objects maintain their rotational state unless disturbed by torque.

4. State the formula for the moment of inertia of a solid cylinder.

   $I = \frac{1}{2} m r^2$. Half the value for solid cylinder compared to hoop.

5. Determine if a system with equal but opposite torques is in equilibrium.

   Yes, it is in rotational equilibrium. Equal opposing torques cancel for zero net torque.

6. How is rotational equilibrium achieved?

   When total clockwise torques equal total counterclockwise torques. Opposing torques must balance for no rotation.

7. What is torque?

   Torque is the rotational equivalent of force, $\tau = r \times F$. Force applied at a distance creates rotational tendency.

8. Which factor does not affect rotational inertia?

   The velocity of the object. Inertia depends on mass and distance, not velocity.

9. Calculate net torque: two forces of $5 \text{N}$ at $r=0.2 \text{m}$ and $r=0.3 \text{m}$, opposite directions.

   $\tau_{net} = 5 \times 0.2 - 5 \times 0.3 = -0.5 \text{Nm}$. Subtract opposing torques to find net effect.

10. Find the torque for $F = 20N$, $r = 0.3m$, $\theta = 90^\text{o}$.

    $\tau = 0.3 \times 20 \times \text{sin}(90) = 6 \text{Nm}$. Perpendicular force gives maximum torque.

11. Calculate net torque: $r = 0.5 \text{m}$, $F = 10 \text{N}$, angle $30^\text{o}$.

    $\tau = 0.5 \times 10 \times \text{sin}(30) = 2.5 \text{Nm}$. Use $\tau = rF\sin\theta$ formula with given values.

12. Identify the factors affecting torque.

    Magnitude of force, distance from pivot, angle of force. All three determine the rotational effect.

13. Identify the unit of torque in the SI system.

    The unit of torque is the Newton-meter (Nm). Force times distance gives units of $N \cdot m$.

14. What happens when net torque is zero?

    The object remains in its current state of rotational motion. No torque means constant angular velocity.

15. Calculate torque for $F = 50 \text{N}$, $r = 0.4 \text{m}$, $\theta = 45^\text{o}$.

    $\tau = 0.4 \times 50 \times \text{sin}(45) = 14.14 \text{Nm}$. Use $\tau = rF\sin\theta$ with $45°$ angle.

16. What is the principle of moments?

    Sum of clockwise moments equals sum of counterclockwise moments for equilibrium. Fundamental rule for rotational equilibrium.

17. What is the significance of the line of action of a force?

    Determines the torque produced by the force. Force direction affects torque magnitude and direction.

18. What is the result of balanced torques on an object?

    The object is in rotational equilibrium. No net torque means no angular acceleration.

19. What is the conservation of angular momentum?

    If no net external torque acts on a system, its angular momentum is conserved. No external torque means angular momentum stays constant.

20. Define the center of mass in the context of rotational motion.

    The point where the mass of a system is concentrated. Balance point where total mass appears concentrated.

21. State the formula for torque as a vector.

    $\tau = \textbf{r} \times \textbf{F}$. Cross product of position and force vectors.

22. How does distribution of mass affect rotational inertia?

    Further mass distribution from the axis increases inertia. Mass farther from axis increases resistance to rotation.

23. State the formula for the moment of inertia of a thin rod about its center.

    $I = \frac{1}{12} m L^2$. Inertia formula for rod rotating about its center.

24. What does the symbol $\tau$ represent in rotational dynamics?

    $\tau$ represents torque. Greek letter tau denotes rotational force.

25. Determine the equilibrium condition for a seesaw.

    The sum of clockwise torques equals the sum of counterclockwise torques. Balanced moments create no net rotation.

26. What is the effect of increasing moment of inertia?

    Decreases angular acceleration for the same torque. Higher inertia requires more torque for same acceleration.

27. Find the moment of inertia for a hoop of mass $m$ and radius $r$.

    $I = m r^2$. All mass is at distance $r$ from the center.

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