Applying Newton’s Laws Chapter 5 Applying Newton’s Laws

Use Newton’s 1st law for bodies in equilibrium (statics) Goals for Chapter 5 Use Newton’s 1st law for bodies in equilibrium (statics) Use Newton’s 2nd law for accelerating bodies (dynamics) Study types of friction & fluid resistance Solve circular motion problems

Using Newton’s First Law when forces are in equilibrium A body is in equilibrium when it is at rest or moving with constant velocity in an inertial frame of reference. Follow Problem-Solving Strategy 5.1.

Using Newton’s First Law when forces are in equilibrium VISUALIZE (create a coordinate system; decide what is happening?) SKETCH FREE-BODY DIAGRAM Isolate one point/body/object Show all forces in that coordinate system ON that body (not acting by that body!) LABEL all forces clearly, consistently Normal forces from surfaces Friction forces from surfaces Tension Forces from ropes Contact forces from other objects Weight from gravity

Using Newton’s First Law when forces are in equilibrium BREAK ALL applied forces into components based on your coordinate system. Apply Newton’s Laws to like components ONLY SFx = max; SFy = may

One-dimensional equilibrium: Tension in a massless rope A gymnast hangs from the end of a massless rope. Example 5.1 (mg = 50 kg; what is weight & force on rope?)

One-dimensional equilibrium: Tension in a rope with mass What is the tension in the previous example if the rope has mass? (Example 5.2 weight of rope = 120 N)

Two-dimensional equilibrium A car engine hangs from several chains. Example 5.3 (weight of car engine = w; ignore chain weights)

A car on an inclined plane An car rests on a slanted ramp. Example 5.4 (car of weight w)

A car on an inclined plane Coordinate system choice #1: (y) parallel to slope & (x) perpendicular to slope y x a b

A car on an inclined plane Coordinate system choice #1: (y) parallel to slope & (x) perpendicular to slope y T (in x) x N (in y) W (in x & y) a b

A car on an inclined plane Coordinate system choice #2: (y) perpendicular to ground (x) parallel to ground N in both x & y! T (in x & y!) y a W (in y only) a x

A car on an inclined plane Coordinate system choice #2: (y) perpendicular to ground (x) parallel to ground N T y Ny Tx a Tx b Nx a W a x

Example 5.5: Bodies connected by a cable and pulley Cart connected to bucket by cable passing over pulley. Initially, assume pulley is massless and frictionless! Pulleys REDIRECT force – they don’t amplify or reduce. Tension in rope pulls upwards along slope SAME tension in rope pulls upwards on bucket

Example 5.5: Bodies connected by a cable and pulley Draw separate free-body diagrams for the bucket and the cart.

A note on free-body diagrams Only the force of gravity acts on the falling apple. ma does not belong in a free-body diagram! It is the SUM of all the forces you find! 

Ex 5.6 Straight-line motion with constant force Wind exerts a constant horizontal force on the boat. 4.0 s after release, v = 6.0 m/s; mass = 200 kg.

Ex 5.6 Straight-line motion with constant force Wind exerts a constant horizontal force on the boat. 4.0 s after release, v = 6.0 m/s; mass = 200 kg. Find W, the force of the wind!

Example 5.7: Straight-line motion with friction For the ice boat in the previous example, a constant horizontal friction force of 100 N opposes its motion What constant force needed by wind to create the same acceleration (a = + 1.5 m/s/s)?

Example 5.7: Straight-line motion with friction For the ice boat in the previous example, a constant horizontal friction force now opposes its motion (100N); what constant force needed by wind to create the same acceleration (a = + 1.5 m/s/s)?

Example 5.8: Tension in an elevator cable Elevator (800 kg) is moving downward @ 10 m/s but slowing to a stop over 25.0 m. What is the tension in the supporting cable?

Example 5.8: Tension in an elevator cable Elevator (800 kg) is moving downward @ 10 m/s but slowing to a stop over 25.0 m. What is the tension in the supporting cable?

Example 5.8: Tension in an elevator cable Compare Tension to Weight while elevator slows?

Example 5.8: Tension in an elevator cable Compare Tension to Weight while elevator slows? What if elevator was accelerating upwards at same rate?

Example 5.8: Tension in an elevator cable What if elevator was accelerating upwards at same rate? Same Free Body Diagram! Same result!

Ex 5.9 Apparent weight in an accelerating elevator A woman inside the elevator of the previous example is standing on a scale. How will the acceleration of the elevator affect the scale reading?

Ex 5.9 Apparent weight in an accelerating elevator A woman inside the elevator of the previous example is standing on a scale. How will the acceleration of the elevator affect the scale reading?

Ex 5.9 Apparent weight in an accelerating elevator What if she was accelerating downward, rather than slowing? Increasing speed down?

Acceleration down a hill What is the acceleration of a toboggan sliding down a friction-free slope?

Acceleration down a hill What is the acceleration of a toboggan sliding down a friction-free slope? Follow Example 5.10.

Two common free-body diagram errors The normal force must be perpendicular to the surface. There is no separate “ma force.” 

Two bodies with the same acceleration We can treat the milk carton and tray as separate bodies, or we can treat them as a single composite body.

Two bodies with the same acceleration Push a 1.00 kg food tray with constant 9 N force, no friction. What is acceleration of tray and force of tray on the carton of milk?

Two bodies with the same magnitude of acceleration The glider on the air track and the falling weight move in different directions, but their accelerations have the same magnitude and relative direction (both increasing, or both decreasing)

Two bodies with the same magnitude of acceleration What is the tension T, and the acceleration a, of the system?

Frictional forces When a body rests or slides on a surface, the friction force is parallel to the surface. Friction between two surfaces arises from interactions between molecules on the surfaces.

Kinetic and static friction Kinetic friction acts when a body slides over a surface. The kinetic friction force is fk = µkn. Static friction acts when there is no relative motion between bodies. The static friction force can vary between zero and its maximum value: fs ≤ µsn.

Static friction followed by kinetic friction Before the box slides, static friction acts. But once it starts to slide, kinetic friction acts.

Some approximate coefficients of friction

Friction in horizontal motion – example 5.13 Move a 500-N crate across a floor with friction by pulling with a force of 230 N. Initially, pull harder to get it going; later pull easier (at 200N once it is going). What are ms? mk?

Friction in horizontal motion Before the crate moves, static friction acts on it.

Friction in horizontal motion After it starts to move, kinetic friction acts.

Static friction can be less than the maximum Static friction only has its maximum value just before the box “breaks loose” and starts to slide. Force Force builds in time to maximum value, then object starts moving & slipping Time

Pulling a crate at an angle The angle of the pull affects the normal force, which in turn affects the friction force. Follow Example 5.15.

Motion on a slope having friction – ex 5.16 Consider a toboggan going down a slope at constant speed. What is m? Now consider same toboggan on steeper hill, so it is now accelerating. What is a?

Fluid resistance and terminal speed Fluid resistance on a body depends on the speed of the body. Resistance can depend upon v or v2 and upon the shape moving through the fluid. Fresistance = -kv or - Dv2 These will result in different terminal speeds

Fluid resistance and terminal speed A falling body reaches its terminal speed when resisting force equals weight of the body. If F = -kv for a falling body, vterminal = mg/k If F = - Dv2 for a falling body, vterminal = (mg/D)½

Dynamics of circular motion If something is in uniform circular motion, both its acceleration and net force on it are directed toward center of circle. The net force on the particle is Fnet = mv2/R, always towards the center.

Using Newton’s First Law when forces are in equilibrium IF you see uniform circular motion… r v

Using Newton’s First Law when forces are in equilibrium IF you see uniform circular motion… THEN remember centripetal force is NOT another force – it is the SUM of one or more forces already present! SFx = mv2/R NOT mv2/R + T – mg = ma r v

Avoid using “centrifugal force” Figure (a) shows the correct free-body diagram for a body in uniform circular motion. Figure (b) shows a common error. In an inertial frame of reference, there is no such thing as “centrifugal force.”

Force in uniform circular motion A 25 kg sled on frictionless ice is kept in uniform circular motion by a 5.00 m rope at 5 rev/minute. What is the force?

Force in uniform circular motion A 25 kg sled on frictionless ice is kept in uniform circular motion by a 5.00 m rope at 5 rev/minute. What is the force?

What if the string breaks? If the string breaks, no net force acts on the ball, so it obeys Newton’s first law and moves in a straight line.

A conical pendulum – Ex 5.20 A bob at the end of a wire moves in a horizontal circle with constant speed.

A car rounds a flat curve A car rounds a flat unbanked curve. What is its maximum speed?

A car rounds a flat curve A car rounds a flat unbanked curve. What is its maximum speed?

A car rounds a banked curve At what angle should a curve be banked so a car can make the turn even with no friction?

A car rounds a banked curve At what angle should a curve be banked so a car can make the turn even with no friction?

Uniform motion in a vertical circle – Ex. 5.23 A person on a Ferris wheel moves in a vertical circle at constant speed. What are forces on person at top and bottom?

Uniform motion in a vertical circle – Ex. 5.23 A person on a Ferris wheel moves in a vertical circle.

The fundamental forces of nature According to current understanding, all forces are expressions of four distinct fundamental forces: gravitational interactions electromagnetic interactions strong interaction weak interaction Physicists have taken steps to unify all interactions into a theory of everything.