FORCES IN TWO DIMENSIONS

EQUILIBRIUM & THE EQUILIBRANT

§ An object is in equilibrium when the net force on it is zero§ Object is motionless or moves with a

constant velocity§ Equilibrant: a single, additional force that is

exerted on an object to produce equilibrium§ To find the equilibrant: find the sum of the 2

forces exerted on the object

What is the relationship between the equilibrant and the resultant vector?§ Equilibrant is the same magnitude

but opposite direction as the sum of all the forces acting on an object

§ One way of determining the net force involves vector resolution (vector components)§ Recall:

§ Ax = Acosθ§ Ay = Asinθ§ R2 = Ax

2 + Ay2

§ θ = tan-1(y/x)

ANALYZE A HANGING SIGN

§ Common example of object in static equilibrium are hanging signs

§ Ex: A sign is held up by two ropes with a tension of 50 N and the angle that each cable makes with the horizontal is known to be 30°. What is the weight of the sign?1. Determine the components of the

tensions2. Sum of the two Ay component is equal

to the weight (mg) of the sign

25 N + 25 N = 50 NWeight of sign = 50 N

YOU TRY!

A picture frame hanging on a wall is held by two ropes each pulling with a force of 30 N with an angle of 45° with respect to the horizontal. What is the weight of the picture frame?

§ 42.4 NWhat is the picture frame’s mass?§ 4.33 kg

Two ropes pull on a ring. One exerts a 62 N force at 30.0°, the other a 62 N force at 60.0°. What is the resultant net force on the ring?*

§ 120 N

How About This One?§ A sign is symmetrically hung by two ropes. The sign is known to have

a mass of 5 kg and the angle between the two ropes is 100°. What is the tension on each cable?§ 38.1 N

MOTION ALONG AN INCLINED PLANE

Which direction does the gravitational force on the object point?§Toward the center of the Earth in a downward

direction; Fg = mg

In which direction is normal force?

§ Perpendicular to the surface

You can break up Fg into components (Fgx, Fgy)§ y-component = what is pushing the object into

the surface§ x-component = force that is trying the

accelerate the object along the incline

Fg

θ Fgy

Fgx

FN

Fgx

§ If Fgy tells us what is pushing the object into the surface, what is the relationship between Fgy and FN?§ They are equal in magnitude

Fg

θ Fgy

Fgx

FN

Fgy = Fg cos θ Fgx = Fg sin θ

θ

It’s important to choose your coordinate plane (to indicate + and – axis)

+x

+y

These angles

are the same (aka F⟂ or Fperp)(aka FII or Fpara)

SAMPLE PROBLEMA 562 N crate is resting on a plane inclined 30.0 above the horizontal. Find the components of the crate’s weight that are parallel and perpendicular to the plane.Given:§ Fg = 562 N§ θ = 30.0°Unknown:§ Fgx = ?§ Fgy = ?

Equation:§ Fgy = Fg cos θ § Fgx = Fg sin θSubstitute & Solve:§ Fgy = Fg cos θ = 562 cos 30° = 487 N

= - 487 N § Fgx = Fg sin θ = 562 sin 30° = 281 N

= - 281 N

Fgx and Fgy are negative because they point in directions opposite to the positive axes

YOU TRY!

§ Scott and Becca are moving a folding table out of the sunlight. A cup of lemonade with a mass of 0.44 kg is on the table. Scott lifts his end of the table before Becca does, and as a result, the table makes an angle of 15.0° with the horizontal. Find the components of the cup's weight that are parallel and perpendicular to plane of table. § Fgx (parallel) = 1.1N ; Fgy (perpendicular) = 4.2 N

§ Fernando, who has a mass of 43.0 kg, slides down the banister at his grandparents’ house. If the banister makes an angle of 35.0° with the horizontal, what is the normal force between Fernando and the banister?§ 345 N

Consider the crate in the incline in the Sample Problem.Calculate the magnitude of the acceleration.§ 4.90 m/s2

After 4.00 s, how fast would the trunk be moving?§ 19.6 m/s