1. Energy and Momentum of Rotating Systems

1.1 Conservation of Angular Momentum

1.1.1 Conservation of Angular Momentum

1.1.2 Angular Momentum of a System

1.2 Rotational Kinetic Energy, Torque & Work

1.2.1 Rotational Kinetic Energy

1.2.2 Torque & Work

1.3 Examples of Rotational Systems

1.3.1 Rolling

1.3.2 Motion of Orbiting Satellites

1.3.3 Escape Velocity

1.4 Angular Momentum & Angular Impulse

1.4.1 Angular Momentum

1.4.2 Angular Impulse

1.4.3 Impulse–Momentum Theorem in Rotational Form

1.4.4 Angular Impulse Graphs

2. Force and Translational Dynamics

2.1 Gravitational Force

2.1.1 Gravitational Field Strength Equation

2.1.2 Weight

2.1.3 Apparent Weight

2.1.4 Newton's Law of Gravitation

2.1.5 Inertial vs Gravitational Mass

2.1.6 Gravitational Force

2.1.7 Gravitational Field

2.1.8 Acceleration Due to Gravity

2.2 Newton's Laws

2.2.1 Newton's Second Law

2.2.2 Newton's Third Law

2.2.3 Newton's First Law

2.3 Systems & Center of Mass

2.3.1 Describing Systems

2.3.2 Center of Mass

2.4 Kinetic & Static Friction

2.4.1 Static Friction Force Formula

2.4.2 Slipping & Sliding

2.4.3 Kinetic Friction

2.4.4 Kinetic Friction Force Equation

2.4.5 Static Friction

2.5 Circular Motion

2.5.1 Kepler's Third Law

2.5.2 Uniform Circular Motion

2.5.3 Acceleration in Circular Motion

2.5.4 Circular Motion in a Vertical Loop

2.5.5 Circular Motion Examples

2.6 Tension

2.6.1 Tension in Ideal & Non-Ideal Strings

2.7 Spring Forces

2.7.1 Hooke's Law

2.8 Forces & Free-Body Diagrams

2.8.1 Free Body Diagrams

2.8.2 Forces as Interactions

2.8.3 Net Force

2.8.4 Translational Equilibrium

3. Fluids

3.1 Fluid Dynamics

3.1.1 Examples of Fluid Dynamics in Real Life

3.1.2 Analyzing Energy Conservation in Fluids

3.1.3 Bernoulli’s Principle and Its Applications

3.2 Fluid Mechanics

3.2.1 Fluid Velocity

3.2.2 Buoyant Force

3.2.3 Fluid Flow Rates

3.2.4 Continuity Equation

3.2.5 Bernoulli’s Equation

3.2.6 Torricelli’s Theorem

3.3 Fluids and Forces

3.3.1 Fluids in Motion and Newton’s Laws

3.3.2 Interactions Between Fluids and Solids

3.3.3 Analyzing Forces in Fluid Systems

3.3.4 Real-World Examples of Fluid Forces

3.4 Density & Pressure

3.4.1 Fluid Properties

3.4.2 Definition of Pressure

3.4.3 Fluid Pressure Equations

4. Work, Energy and Power

4.1 Translational Kinetic Energy & Potential Energy

4.1.1 Gravitational Potential Energy Between Objects

4.1.2 Translational Kinetic Energy

4.1.3 Potential Energy

4.1.4 Elastic Potential Energy

4.1.5 Gravitational Potential Energy Near a Surface

4.2 Work-Energy Theorem

4.2.1 Work-Energy Theorem

4.3 Work

4.3.1 Defining Work & Work Done

4.3.2 Calculating Work Done

4.4 Conservation of Energy

4.4.1 Conservation of Energy

4.5 Power

4.5.1 Calculating Power

4.5.2 Instantaneous Power

5. Torque and Rotational Dynamics

5.1 Newton’s First & Second Law in Rotational Form

5.1.1 Newton’s Second Law in Rotational Form

5.1.2 Rotational Equilibrium

5.1.3 Newton’s First Law in Rotational Form

5.2 Rotational Kinematics

5.2.1 Rotational Kinematics Graphs

5.2.2 Rotating Systems

5.2.3 Angular Displacement

5.2.4 Angular Velocity

5.2.5 Angular Acceleration

5.2.6 Connecting Linear & Rotational Motion

5.2.7 Rotational Kinematics Equations

5.3 Torque

5.3.1 Defining Torque

5.3.2 Force Diagrams & Torque

5.4 Rotational Inertia

5.4.1 Rotational Inertia

5.4.2 Parallel Axis Theorem

6. Kinematics

6.1 Displacement, Velocity & Acceleration

6.1.1 Velocity

6.1.2 Acceleration

6.1.3 Displacement

6.2 Representing Motion

6.2.1 Kinematic Equations

6.2.2 Motion Graphs

6.2.3 Reference Frames & Relative Motion

6.3 Scalars & Vectors

6.3.1 Scalar & Vector Quantities

6.3.2 Combining Vectors

6.4 Vectors & Motion

6.4.1 Vectors & Motion in Two Dimensions

6.4.2 Projectile Motion

7. Oscillations

7.1 Representing and Analyzing SHM

7.1.1 Features of SHM

7.1.2 Graphical Representation of SHM

7.2 Energy of Simple Harmonic Oscillators

7.2.1 Total Energy of SHM

7.2.2 Kinetic Energy & Potential Energy of SHM

7.3 Simple Harmonic Motion

7.3.1 Defining Simple Harmonic Motion

7.3.2 Frequency & Period of SHM

8. Linear Momentum

8.1 Conservation of Linear Momentum

8.1.1 Elastic & Inelastic Collisions

8.1.2 The Principle of Conservation of Momentum

8.1.3 Momentum of a System

8.1.4 Momentum in Collisions

8.1.5 Momentum in Explosions

8.2 Linear Momentum & Impulse

8.2.1 Impulse Equation

8.2.2 Impulse–Momentum Theorem

8.2.3 Impulse Graphs

8.2.4 Linear Momentum

8.2.5 Rate of Change of Momentum

Defining Torque

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Defining Torque

Introduction

Torque is a fundamental concept in physics, particularly in the study of rotational dynamics. It quantifies the rotational effect produced by a force applied at a distance from an axis of rotation. Understanding torque is essential for Collegeboard AP Physics 1 students as it forms the basis for analyzing various mechanical systems, from simple levers to complex machinery. Mastery of torque concepts enables students to solve problems related to equilibrium, rotational motion, and the mechanics of everyday objects.

Key Concepts

Definition of Torque

Torque, often represented by the Greek letter τ, is a measure of the rotational force applied to an object. It determines how much a force acting on an object causes that object to rotate. The magnitude of torque depends on two factors: the magnitude of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force.

Formula for Torque

The quantitative relationship for torque is expressed as: $$ \tau = r \cdot F \cdot \sin(\theta) $$ where:

τ is the torque.
r is the lever arm or the perpendicular distance from the axis of rotation to the line of action of the force.
F is the magnitude of the applied force.
θ is the angle between the force vector and the lever arm.

In cases where the force is applied perpendicular to the lever arm, $\sin(\theta) = 1$, simplifying the equation to: $$ \tau = r \cdot F $$

Direction of Torque

Torque is a vector quantity, meaning it has both magnitude and direction. The direction of torque is determined by the right-hand rule:

Point your fingers in the direction of the lever arm (from the axis to the point of force application).
Curl your fingers in the direction of the applied force.
Your thumb points in the direction of the torque vector.

A positive torque causes counterclockwise rotation, while a negative torque causes clockwise rotation.

Equilibrium and Torque

An object is in rotational equilibrium when the net torque acting on it is zero. This condition is essential for objects at rest or moving with constant angular velocity. Mathematically, this is expressed as: $$ \sum \tau = 0 $$ For rotational equilibrium:

Clockwise torques must balance counterclockwise torques.
No net rotational acceleration occurs.

Applications of Torque

Torque is pivotal in various applications, including:

Levers: Torque explains how levers can magnify force, making tasks easier.
Automobiles: Engine torque affects vehicle acceleration and performance.
Engineering: Torque calculations are crucial in designing mechanical systems and structures.

Understanding torque allows engineers and physicists to design efficient systems and solve practical problems involving rotational forces.

Moment of Inertia and Torque

The moment of inertia (I) quantifies an object's resistance to changes in its rotational motion. It plays a crucial role in the relationship between torque and angular acceleration ($\alpha$): $$ \tau = I \cdot \alpha $$ This equation parallels Newton's second law for linear motion ($F = m \cdot a$), connecting torque to the angular counterpart of force and acceleration.

Torque in Rotational Dynamics

In rotational dynamics, torque is essential for analyzing the motion of rotating objects. It helps determine angular acceleration and understand how forces cause objects to spin or change their rotational speed. By applying torque equations, one can solve complex problems involving multiple forces and rotational axes.

Torque and Energy

Torque is intrinsically linked to rotational work and kinetic energy. The work done by a torque is given by: $$ W = \tau \cdot \theta $$ where $\theta$ is the angular displacement. Additionally, the rotational kinetic energy (K) of an object is: $$ K = \frac{1}{2} I \cdot \omega^2 $$ where $\omega$ is the angular velocity. These relationships are vital for understanding energy conservation in rotational systems.

Comparison Table

Aspect	Torque	Force
Definition	Measure of rotational influence of a force.	Push or pull acting on an object.
Formula	$\tau = r \cdot F \cdot \sin(\theta)$	$F = m \cdot a$
Units	Newton-meter (N.m)	Newton (N)
Direction	Determined by right-hand rule (vector)	Linear direction of application
Effect	Causes rotational motion	Causes linear motion

Summary and Key Takeaways

Torque quantifies the rotational effect of a force applied at a distance from an axis.
The torque formula considers force magnitude, lever arm length, and force application angle.
Torque direction is determined using the right-hand rule, indicating rotational direction.
Rotational equilibrium occurs when the sum of all torques equals zero.
Understanding torque is essential for analyzing rotational dynamics and engineering applications.

Examiner Tip

Tips

Mnemonic for Torque Direction: Use the "Right-Hand Grip" mnemonic: curl your right hand's fingers in the direction of force application, and your thumb will point in the torque's direction.

Remember the Torque Formula: Think of torque as "r times F times sine theta" ($\tau = r \cdot F \cdot \sin(\theta)$) to always consider the perpendicular component of the force.

Practice with Equilibrium: When dealing with rotational equilibrium, list all torques and set their sum to zero to solve for unknowns efficiently.

Did You Know

Did you know that torque plays a critical role in the design of wind turbines? Engineers calculate the torque generated by wind forces to ensure that the blades spin efficiently and harness maximum energy. Additionally, torque is fundamental in sports, such as when a baseball pitcher applies torque to the ball to achieve the desired spin and trajectory.

Common Mistakes

Mistake 1: Ignoring the angle $\theta$ between the force and the lever arm, leading to incorrect torque calculations.
Incorrect: $\tau = r \cdot F$
Correct: $\tau = r \cdot F \cdot \sin(\theta)$

Mistake 2: Confusing torque direction with force direction, resulting in errors when applying the right-hand rule.
Incorrect: Assuming torque direction is the same as the applied force.
Correct: Use the right-hand rule to determine torque direction based on the force's application.

Mistake 3: Overlooking units of torque, which can lead to incorrect problem-solving steps.
Incorrect: Mixing up Newtons and Newton-meters.
Correct: Always use Newton-meters (N·m) for torque measurements.

FAQ

What is torque?

Torque is a measure of the rotational force applied to an object, determining how much a force causes an object to rotate around an axis.

How is torque calculated?

Torque is calculated using the formula $\tau = r \cdot F \cdot \sin(\theta)$, where $r$ is the lever arm, $F$ is the force applied, and $\theta$ is the angle between the force and the lever arm.

What are the units of torque?

The unit of torque is the Newton-meter (N·m).

How does torque affect rotational equilibrium?

An object is in rotational equilibrium when the sum of all torques acting on it is zero, meaning there is no net rotational acceleration.

What is the relationship between torque and angular acceleration?

Torque is directly proportional to angular acceleration, as expressed by the equation $\tau = I \cdot \alpha$, where $I$ is the moment of inertia and $\alpha$ is the angular acceleration.

Can torque be positive or negative?

Yes, torque can be positive or negative depending on the direction of rotation it causes. Positive torque typically causes counterclockwise rotation, while negative torque causes clockwise rotation.