Tension

Tension

• Tension is the force transmitted through a rope or cable when pulled tight.

• In static equilibrium, tension is uniform along the rope.

• Tension isn't the sum of pulling forces but rather the force at a point.

Tug of War Example

• Tension equals the force applied by one side, not the total of both sides.

Tension and Weight

• Tension isn't always the weight of the supported object; pulleys can change the force distribution.

Forces on Masses Connected by a Rope

• Consider applied force and tension force when analyzing connected masses.

Three Ropes Example

• Resolve tensions into x- and y-components using a free body diagram.

• Use $<br>\Sigma F<em>x = 0$ $\Sigma F</em>y = 0$
  to solve for unknown tensions.

Two Blocks Accelerated by a Force

• Use $F = ma$ to find system acceleration.

• Calculate tension using one block's mass and the system's acceleration.

Massless String Approximation

• Strings are assumed massless, transmitting force unchanged.

Pulleys

• Pulleys redirect tension force without changing its magnitude (ideal conditions).

Pulleys: Basic Usage

• Basic pulley usage applies the same force as lifting directly.

Pulleys: Advantageous Usage

• Pulleys can split force, reducing the required pull.

Example with Multiple Pulleys

• Calculate tensions in each rope segment, considering force splitting.

Horizontal Pulleys

• Simplify force diagrams by changing force direction.

Example 1: Dynamic Equilibrium

• Relate tension to friction to maintain equilibrium.

Example 2: Acceleration with Friction

• Calculate acceleration considering friction.

Atwood Machine

• Relates tensions and masses to acceleration

Multiple Coordinate Systems

• Use separate coordinate systems for each mass in a system.

Pulley on an Incline

• Combines pulley systems with inclined planes.

Example: Mass on an Incline

• Determines minimum mass to prevent sliding using static friction.

Example: Sliding Mass on an Incline

• Calculates acceleration considering kinetic friction.

Example: Jane and the Pulley System

• Analyzes tensions in a multi-pulley lifting system.

Example: Blocks and Friction

• Finds maximum mass for equilibrium considering static friction.

Centripetal Acceleration

• $a_c = \frac{v^2}{r} = \omega^2 r$

Coordinates

• X-y coordinates not ideal for circular motion.

rtz Coordinates

• Use radial and tangential axes.

Uniform Circular Motion

• Constant speed, centripetal acceleration.

Centripetal Force

• Net force towards the circle's center.

Centripetal Force is Not a Separate Force

• Describes direction, not a distinct force.

Uniform Circular Motion

• $<br>\Sigma F<em>r = ma</em>r = m \frac{v^2}{r} = m \omega^2 r$
  $<br>\Sigma F<em>t = 0$ $\Sigma F</em>z = 0$

Example: Car Turning a Corner

• Max speed without sliding: $v = \sqrt{\mu_s gr}$.

Banking Angle

• Angle for frictionless curve: $<br>\theta = \tan^{-1} \left(\frac{v^2}{gR}\right)$

Newton’s 2nd Law

• Relates forces to centripetal acceleration in banked turns.

Centripetal Acceleration

• Provided by friction, tension, or gravity.

Example: Rock Whirling in a Circle

• Find angle using: $<br>\theta = \tan^{-1} \left(\frac{g \tau^2}{4 \pi^2 r}\right)$

The Real Earth vs. Flat-Earth Approximation

• Earth is locally flat.

Orbital Motion

• Free fall moving too fast to hit the ground.

Uniform Circular Motion

• Net force causes centripetal acceleration.

Object in Orbit

• Balance of velocity and centripetal acceleration.

Newton’s Law of Gravity

• Describes force between masses.

Orbital Velocity

• $v = \sqrt{G \frac{M}{R}}$

• $a = \frac{v^2}{R} = G \frac{M}{R^2}$

Example: Astronauts in Orbit

• Calculate gravitational acceleration at orbit altitude.

Orbital Velocity

• Velocity for constant free-fall.

Example: Satellite Orbital Velocity

• Compute satellite's orbital velocity.

Example: Planet Mass

• Determine planet's mass from satellite's orbit.

Orbital Motion

• $T = \sqrt{\frac{4 \pi^2 r^3}{GM}}$

Kepler’s Third Law

• $T^2 = \left(\frac{4 \pi^2}{GM}\right) r^3$

Water bucket

• Roller Coaster Loop-the-Loop
  Minimum speed
  Maximum speed

Loop

• At the top of the loop: Critical speed - the slowest speed at which the car can complete the loop. i.e. where the track provides no normal force and gravity alone provides the centripetal acceleration to keep the car on the track. Critical speed vt