#AP Physics C: Mechanics - Torque & Rotational Statics Study Guide 🚀

Hey there, future physics pro! Let's get you prepped and confident for the exam. This guide is designed to be your go-to resource, especially the night before the test. We'll break down everything you need to know about torque and rotational statics, making sure it all clicks into place. Let's dive in!

#1. Torque: The Rotational Force

#What is Torque? 🤔

Torque ($\tau$) is the rotational equivalent of force. It's what causes an object to rotate around an axis. Think of it as the 'twist' that makes things spin. It's measured in Newton-meters (N·m), which is equivalent to (kg·m²/s²).

#Calculating Torque

The formula for torque involves a cross product:

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Where:

• $\vec{\tau}$ is the torque vector
• $\vec{r}$ is the position vector from the axis of rotation to the point where the force is applied (also known as the lever arm)
• $\vec{F}$ is the force vector

Visually, it looks like this:

• Magnitude of Torque: $\tau = rF\sin(\theta)$, where $\theta$ is the angle between the force and the lever arm. When force is perpendicular to the lever arm, the equation simplifies to $\tau = rF$.

• Direction of Torque: Use the right-hand rule! Point your fingers in the direction of $\vec{r}$, curl them towards $\vec{F}$, and your thumb points in the direction of the torque.

#Torque in Action

Here's a real-world example:

#2. Conditions for Equilibrium

For an object to be in equilibrium, it must satisfy both of these conditions:

#First Condition: Translational Equilibrium

This means the net force acting on the object is zero. In simpler terms, the object is not accelerating linearly. Mathematically:

$$\sum \vec{F} = 0$$

#Second Condition: Rotational Equilibrium

This means the net torque acting on the object is zero. The object is not accelerating angularly. Mathematically:

$$\sum \vec{\tau} = 0$$

#Key Ideas:

• Clockwise vs. Counterclockwise: When solving problems, it's crucial to assign a sign convention (e.g., clockwise torques are negative, counterclockwise are positive).
• Pivot Point: The choice of pivot point is arbitrary. However, choosing a point where an unknown force acts can simplify calculations by eliminating that force's torque.

#3. Moment of Inertia: Resistance to Rotation

#What is Moment of Inertia? 🤔

Moment of Inertia (I), also known as rotational inertia, is an object's resistance to changes in its rotational motion. It depends on how the mass is distributed around the axis of rotation. The further the mass is from the axis, the greater the moment of inertia.

#Calculating Moment of Inertia

The general formula is:

$$I = \int r^2 dm$$

Don't worry, you won't have to do this integration on the AP exam! You'll be given the moment of inertia for common shapes or will be asked to use the parallel axis theorem.

#Common Shapes

Here are some common shapes and their moments of inertia:

#Key Ideas:

• Scalar Quantity: Moment of inertia is a scalar quantity. It only has magnitude, not direction.
• Mass Distribution: The distribution of mass is crucial. Objects with mass further from the axis of rotation have higher moments of inertia.

#4. Parallel Axis Theorem: Shifting the Axis

#What is the Parallel Axis Theorem? 🤔

The parallel axis theorem helps you calculate the moment of inertia about an axis parallel to one that passes through the center of mass. It's super handy when the axis of rotation isn't at the center of mass.

#The Formula

$$I = I_{cm} + Mh^2$$

Where:

• $I$ is the moment of inertia about the new axis.
• $I_{cm}$ is the moment of inertia about the center of mass.
• $M$ is the total mass of the object.
• $h$ is the distance between the new axis and the axis through the center of mass.

#How to Use It:

1. Identify the object and the new axis of rotation.
2. Find the moment of inertia about the center of mass ($I_{cm}$). You'll either be given this or need to know it for a common shape.
3. Determine the distance ($h$) between the two axes.
4. Plug the values into the formula and solve for $I$.

#5. Final Exam Focus

#High-Value Topics

• Torque Calculations: Practice finding torques given forces and lever arms, especially when the force is not perpendicular.
• Equilibrium Problems: Master setting up equations for both translational and rotational equilibrium. Pay attention to sign conventions.
• Moment of Inertia: Be comfortable with using given formulas and the parallel axis theorem.
• Rotational Dynamics: Understand how torque relates to angular acceleration ($\tau = I\alpha$).

#Common Question Types

• Multiple Choice: Conceptual questions about torque, equilibrium, and moment of inertia. Expect questions that test your understanding of vector nature of torque and how mass distribution affects moment of inertia.
• Free Response: Problems involving complex systems in equilibrium, often requiring you to calculate multiple forces and torques. You might also be asked to derive or use the parallel axis theorem.

#Last-Minute Tips

• Draw Free-Body Diagrams: Always start by drawing a clear free-body diagram, including all forces and their points of application.
• Choose Pivot Points Wisely: Select a pivot point that simplifies your calculations, often at the location of an unknown force.
• Check Units: Ensure your units are consistent throughout the problem. Torque is measured in N·m, and moment of inertia in kg·m².
• Stay Calm: Take a deep breath, read the question carefully, and approach it step-by-step. You've got this!

#6. Practice Questions

Here are some practice questions to get you ready for the exam:

Practice Question

Multiple Choice Questions

1. Two children push on opposite sides of a door during play. Both push horizontally and perpendicular to the door. One child pushes with a force of 17.5 N at a distance of 0.600 m from the hinges, and the second child pushes at a distance of 0.450 m. What force must the second child exert to keep the door from moving? Assume friction is negligible.

   • A) 11.2 N
   • B) 17.5 N
   • C) 23.3 N
   • D) 27.7 N

   Answer:

   

2. A uniform rod of mass M and length L is pivoted at one end. What is the moment of inertia of the rod about this pivot point?

   • A) $1/12 ML^2$
   • B) $1/3 ML^2$
   • C) $1/2 ML^2$
   • D) $ML^2$

   Answer: B

3. A solid sphere and a hollow sphere, both with the same mass and radius, are released from rest at the top of an incline. Which sphere will reach the bottom first?

   • A) The solid sphere
   • B) The hollow sphere
   • C) They will reach the bottom at the same time
   • D) It depends on the angle of the incline

   Answer: A

Free Response Question

A wheel of mass $M$ and radius $R$ is mounted on a horizontal axle. A string is wrapped around the wheel and a block of mass $m$ is attached to the end of the string. The block is released from rest and falls, causing the wheel to rotate. Assume the string does not slip on the wheel.

(a) On the diagram below, draw and label all the forces acting on the block and the wheel.

(b) Derive an expression for the acceleration of the block in terms of $M$, $m$, $R$, and $g$. Assume the moment of inertia of the wheel is $I = \frac{1}{2}MR^2$.

(c) Derive an expression for the tension in the string in terms of $M$, $m$, $R$, and $g$.

(d) If the block falls a distance $h$, what is the angular speed of the wheel?

Answer:

Scoring Breakdown:

• (a) Correct Free Body Diagrams: 2 points (1 for each correct FBD)
• (b) Correct Derivation of Acceleration: 4 points
  • 2 points for correct Newton's second law for the block
  • 1 point for correct torque equation for the wheel
  • 1 point for correct kinematic relationship between linear and angular acceleration
• (c) Correct Derivation of Tension: 2 points
  • 1 point for using the derived acceleration
  • 1 point for correct tension equation
• (d) Correct Derivation of Angular Speed: 2 points
  • 1 point for using energy conservation
  • 1 point for correct final expression

Short Answer Question

A uniform meter stick of mass $M$ is pivoted at the 20 cm mark. A mass of $2M$ is hung at the 80 cm mark. At what position on the meter stick should a mass of $M$ be hung to achieve equilibrium?

Answer: