Chapter 8Chapter 8

StaticsStatics

EquilibriumEquilibrium

An object either at rest or moving An object either at rest or moving with a constant velocity is said to be with a constant velocity is said to be in in equilibriumequilibrium

The net force acting on the object is The net force acting on the object is zero (since the acceleration is zero)zero (since the acceleration is zero)

0F

11stst Condition for Equilibrium Condition for Equilibrium

Net force vanishesNet force vanishes

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MethodMethod

Isolate object of interestIsolate object of interest Draw picture, label all forcesDraw picture, label all forces Choose appropriate coordinate Choose appropriate coordinate

systemsystem Resolve into componentsResolve into components Write equations for equilibriumWrite equations for equilibrium Solve equations for unknownsSolve equations for unknowns

ExampleExample

Mass m on table topMass m on table top 200 lb man hanging from rope200 lb man hanging from rope

Example (ex 2)Example (ex 2)

Find tensions in the ropes.Find tensions in the ropes.

Example 8.4Example 8.4

Two objects are in equilibrium. Find Two objects are in equilibrium. Find the weight of the one on the left and the weight of the one on the left and tensions.tensions.

Force vs. TorqueForce vs. Torque

Forces cause accelerationsForces cause accelerations Torques cause angular accelerationsTorques cause angular accelerations

• rotationrotation Force and torque are relatedForce and torque are related

TorqueTorque

The door is free to rotate about an axis through OThe door is free to rotate about an axis through O There are three factors that determine the There are three factors that determine the

effectiveness of the force in opening the door:effectiveness of the force in opening the door:• The The magnitudemagnitude of the force of the force• The The positionposition of the application of the force of the application of the force• The The angleangle at which the force is applied at which the force is applied

Torque, contTorque, cont

Torque, Torque, , is the tendency of a force to , is the tendency of a force to rotate an object about some axisrotate an object about some axis

is the torqueis the torque F is the forceF is the force

• symbol is the Greek tausymbol is the Greek tau ll is the lever arm is the lever arm

SI unit is NSI unit is N..mm Lever Arm: Perpendicular distance from Lever Arm: Perpendicular distance from

the pivot point to the line of forcethe pivot point to the line of force

Fl

Direction of TorqueDirection of Torque

Torque is a vector quantityTorque is a vector quantity• We will treat only 2-d torque so no We will treat only 2-d torque so no

need for vector notion.need for vector notion.• If the turning tendency of the force is If the turning tendency of the force is

counterclockwise, the torque will be counterclockwise, the torque will be positive (+)positive (+)

• If the turning tendency is clockwise, If the turning tendency is clockwise, the torque will be negative (-)the torque will be negative (-)

Multiple TorquesMultiple Torques

When two or more torques are acting When two or more torques are acting on an object, the torques are addedon an object, the torques are added• with the signswith the signs

If the net torque is zero, the object’s If the net torque is zero, the object’s rate of rotation doesn’t changerate of rotation doesn’t change

General Definition of TorqueGeneral Definition of Torque

The applied force is not always The applied force is not always perpendicular to the position vectorperpendicular to the position vector

The component of the force The component of the force perpendicularperpendicular to the object will cause to the object will cause it to rotateit to rotate

sin

sin

Ll

FL

F is the forceF is the force

L is distance L is distance between pivot and between pivot and point of actionpoint of action

is the angleis the angle

Example (8.20)Example (8.20)

Torque problemTorque problem

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Center of GravityCenter of Gravity

The weight of an object can cause The weight of an object can cause object to pivotobject to pivot

C.G. used as pivot will have no C.G. used as pivot will have no torque due to gravitytorque due to gravity

In finding the torque produced by the In finding the torque produced by the force of gravity, all of the weight of force of gravity, all of the weight of the object can be considered to be the object can be considered to be concentrated at a single point, C.G.concentrated at a single point, C.G.

Center of Gravity of a Uniform Center of Gravity of a Uniform ObjectObject

The center of gravity of a The center of gravity of a homogenous, symmetric body must homogenous, symmetric body must lie on the axis of symmetry.lie on the axis of symmetry.

Often, the center of gravity of such Often, the center of gravity of such an object is the an object is the geometricgeometric center of center of the object.the object.

Experimentally Determining the Experimentally Determining the Center of GravityCenter of Gravity

The wrench is hung freely The wrench is hung freely from two different pivotsfrom two different pivots

The intersection of the The intersection of the lines indicates the center lines indicates the center of gravityof gravity

A rigid object can be A rigid object can be balanced by a single balanced by a single force equal in magnitude force equal in magnitude to its weight as long as to its weight as long as the force is acting the force is acting upward through the upward through the object’s center of gravityobject’s center of gravity

22ndnd Condition for Equilibrium Condition for Equilibrium

No rotationNo rotation Net torque vanishesNet torque vanishes

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ExampleExample

1000 N sign painter on an uniform 1000 N sign painter on an uniform 250N board suspended by two ropes. 250N board suspended by two ropes. The board is 5m long. If the rope can The board is 5m long. If the rope can hold only 900 N, is it safe for him to hold only 900 N, is it safe for him to stand 1m from a rope?stand 1m from a rope?

ExampleExample

600 N worker 600 N worker standing 1.5 m standing 1.5 m from the wall on from the wall on 5m long and 300N 5m long and 300N beam as show. beam as show.

Find tension in the Find tension in the cable and the cable and the force on the beam force on the beam by the wall.by the wall.