PHYS 225 - Mechanics - Lecture 20 - FBD | Tension

Learning Goals

  • Practice solving force and motion problems, focusing on:

    • Inclines

    • Tension forces

Force and Motion on Inclines

  • Inclined Plane Problem: Consider a block of mass $m_1 = 104$ kg pushed at constant velocity up a frictionless ramp at an angle $ heta = 33°$.

    • Positive direction of x-axis: Up the ramp.

    • Positive direction of y-axis: Perpendicular to the ramp.

Clicker Question 9: Weight Components

  • When evaluating the components of weight ($W$):

    • The x-component ($Wx$) and y-component ($Wy$) of weight can be expressed as:

    • Wx=∣W∣imesextcos(heta)W_x = |W| imes ext{cos}( heta)

    • Wy=∣W∣imesextsin(heta)W_y = |W| imes ext{sin}( heta)

Clicker Question 10: Force Components

  • For a horizontal force $F$ acting on the block:

    • The components of $F$ can be expressed as:

    • Fx=∣F∣imesextcos(heta)F_x = |F| imes ext{cos}( heta)

    • Fy=∣F∣imesextsin(heta)F_y = |F| imes ext{sin}( heta)

Tension Forces

  • Definition of Tension:

    • A force related to the stretching of an object, typically through strings or cables.

    • It acts along the string and is pulling at both ends, represented as $F_T$ (tension force).

Real-Life Examples of Tension

  • Tension is prevalent in many mechanical systems, including:

    • Elevators

    • Suspension bridges

    • Pulleys and dyneema cords

Atwood Machine

  • Description: An Atwood machine consists of pulleys, strings, and masses used to demonstrate the principles of tension.

  • Key Characteristics:

    • Tension remains constant throughout an ideal string.

    • Mass and friction of pulleys are negligible for simpler calculations.

  • Ideal vs Non-Ideal Conditions:

    • In ideal conditions, tension remains constant wherever the string transmits the force, regardless of path.

Clicker Questions on Tension and Acceleration

  • Question 1: A cart ($w1$) attached to a bucket ($w2$) on a frictionless ramp accelerates up. Forces involved:

    • Weight of cart ($w_1$)

    • Weight of bucket ($w_2$)

    • Tension ($T$) acting upwards on the cart.

  • Question 2: In a balanced system where all masses ($mA = mB$) are at rest:

    • Rank the tension forces in different setups (A, B, C) using the following options:

    • A. $TA = TB = T_C$

    • B. $TA < TB < T_C$

    • C. $TA = TB < T_C$

Key Notes for Preparation

  • Understand the fundamentals of tension and how it transmits forces in systems.

  • Solve problems involving components of forces on inclines.

  • Familiarize with Atwood machines and their workings to internalize how tension is used in practical applications.

Inclined Plane Problem

  • Block mass: $m_1 = 104$ kg

  • Angle of incline: $\theta = 33°$

  • Positive x-axis: Up the ramp.

  • Positive y-axis: Perpendicular to the ramp.

Weight Components

  • x-component of weight ($Wx$): W</em>x=∣W∣×cos⁡(θ)W</em>x = |W| \times \cos(\theta)

  • y-component of weight ($Wy$): W</em>y=∣W∣×sin⁡(θ)W</em>y = |W| \times \sin(\theta)

Force Components

  • x-component of force ($Fx$): F</em>x=∣F∣×cos⁡(θ)F</em>x = |F| \times \cos(\theta)

  • y-component of force ($Fy$): F</em>y=∣F∣×sin⁡(θ)F</em>y = |F| \times \sin(\theta)

Tension Forces

  • Tension ($F_T$): A force related to the stretching of an object, acts along the string being stretched.

Atwood Machine

  • Tension remains constant throughout an ideal string and mass and friction of pulleys are negligible.

  • In ideal conditions, tension remains constant regardless of the path taken by the string.