AP Physics Slide Set 5.1-Torque

Unit 5: Torque and Rotational Motion

Slide Set Overview

• Introduction to torque and its significance in rotational dynamics.

Rotational Kinematics Review

Key Equations:

1. Displacement:

   • Δx = Δs = r × Δθ Where:

   • Δx: Linear displacement

   • r: Radius (distance from the axis of rotation)

   • Δθ: Angular displacement

2. Velocity:

   • v = r × ω Where:

   • v: Linear velocity

   • ω: Angular velocity

3. Acceleration:

   • a = r × α Where:

   • a: Linear acceleration

   • α: Angular acceleration

Rotational Forces - Torque

Definition of Torque:

• Torque (τ):

  • τ = r × F × sin(θ)Where:

  • τ: Torque

  • r: Distance from the axis of rotation to the point force is applied (lever arm)

  • F: Magnitude of the applied force

  • θ: Angle between the force and lever arm

• Notable difference from linear work equation:

  • W = F × d × cos(θ).

Non-zero Net Torques

• Analogy to Newton's Second Law:

  • Non-zero net forces lead to translational acceleration.

  • Non-zero net torques lead to angular acceleration.

  • Torque is a measure of a force's effectiveness in causing an object to rotate.

Axis of Rotation and Torque Calculation

Force's Effect on Rotation:

• For effective rotation, the force's line of action must not cross the axis of rotation.

• Torque is determined by the angle (θ) and distance (r), impacting lever arm length.

Components of Torque

• Torque's Dependence on Force Components:

  • Only the component of force perpendicular to the lever arm (not passing through the axis) affects the torque.

  • Example: Pushing on the edge of a door does not generate torque as it acts through the axis.

Torque Direction Convention

Positive and Negative Torque:

• Counter-clockwise (CCW) motion is considered positive.

• Clockwise (CW) motion is considered negative.

• Use of the right-hand rule for determining torque direction:

  • Thumb points in the direction of positive torque.

Equilibrium Condition

Equilibrium in Systems:

• For a system to be in equilibrium, the sum of all forces and torques must equal zero:

  • ΣF = 0

  • Στ = 0

• CCW and CW torques must be balanced:

  • τCCW = τCW

Choosing Axis of Rotation

Flexibility in Axis Selection:

• The choice of axis of rotation affects torque calculations but does not favor any specific axis.

• Selecting strategic axes can simplify problems.

Torque Examples

Example of Torque with Fulcrum:

• If the fulcrum's tip is used as the axis:

  • Weight (w1) creates CCW torque (positive).

  • Weight (w2) creates CW torque (negative).

  • Force (F_p) through the AoR produces zero torque.

Using a Cheater Bar

• Increasing Applied Torque:

  • Using a longer lever arm (cheater bar) allows for increased torque without increasing applied force.

  • Caution advised when using heavy machinery to avoid accidents.

Example Problem: Push-Up Torque Analysis

Problem Statement:

• A woman weighing 65 kg in the push-up position. Analyze forces on her hands and feet.

Equilibrium Conditions:

Forces in Equilibrium:

• All forces and torques must balance, represented by:

  • ΣF = 0

  • Στ = 0

Force Calculation:

• Sum of Forces:

  • FH (force on hands) + FF (force on feet) must counterbalance weight:

  • FH + FF - w = 0

  • w = 65 kg * g ≈ 637 N

Choosing Axis for Torque Calculation

Selecting Hands as AoR:

• FH produces no torque when hands are the AoR.

• Simplifying the torque equation:

  • FH + FF = 637 N

Solving for Torque:

Setting up Torque Equation:

• Στ for the woman using hands as AoR:

  • Assuming sin(θ) = 1 for simplicity:

  • FF × 1.5 m - 637 N × 0.5 m = 0

Solving for Forces:

Calculate FF:

• FF × 1.5 m = 318.5 N·m

  • FF = 212 N

Final Force Calculations:

Finding FH:

• FH + FF = 637 N

  • FH = 425 N (637 N - 212 N = 425 N)

Summary of Forces in Equilibrium:

• Final forces determined:

  • FH = 425 N (force on hands)

  • FF = 212 N (force on feet)