AP Physics C Mechanics Unit 2: Force and Translational Dynamics

Newton’s Laws

Newton’s First Law: The Law of Inertia

An object at rest stays at rest unless acted on by an external force. An object in motion continues to travel with constant velocity unless acted on by an external force.

• slowing of objects in everyday experience is due to the force of friction

  • if friction is reduced, rate of slowing is reduced

• if we could remove from an object all external forces, including friction, then the velocity of the object would never change

  • inertia

Newton’s Second Law: The Law of Acceleration

The direction of the acceleration of an object is in the direction of thee net external force acting on it. The acceleration is proportional to the net external force in accordance with F_net = ma where m is the mass of the object. The net force actin on an object also called teh resultant force is the vector sum of all the forces acting on it. F_net = vector sum of F, thus:

sum F = ma

Newton’s Third Law: The Law of Action and Reaction

Forces always occur in equal and opposite pairs. If object A exerts.a force F_AB on object B, an equal but opposite force F_BA is exerted by object B on object A, thus:

F_BA = F_AB

Inertial Reference Frames

• whether an object remains at rest or remains moving with constant velocity depends on the reference frame in which an object is observed

• if you put ball in cupholder of plane, relative to passenger in plane the ball is not moving, but relative to the ground the ball is moving at the same velocity as the plane

• a reference frame accelerating relative to an inertial reference frame is not an inertial reference frame

  • newton’s first law thus gives us the criterion for determining if a reference frame is inertial frame

Inertial Reference Fram: If no forces act on an object, any reference frame with respect to which the acceleration of the object remains zero is an inertial reference frame.

• any reference frame moving with constant velocity relative to an inertial reference frame is also an inertial reference frame

• a reference frame attached to the surface of the earth is not technically an inertial reference frame because of the small acceleration of the surface of the earth due to earth rotation and earth rotation around the sun

  • for good approximation, reference frame attached to the surface of the earth is an inertial reference frame

  • Newton’s first, second and thrip law statements are only valid in inertial reference frames

Force, Mass and Newton’s Second Law

• force is an external influence on an object that caused it to accelerate relative to an inertial reference frame

  • we assume there are no other forces acting

• direction of force is the direction of the acceleration it causes

• magnitude of force is the product of the mass of the object and the magnitude of its acceleration

• objects intrinsically resist being accelerated

• mass is measure of the object’s inertia

• m₂/m₁ = a₁/a₂

• if a force is applied to an object and a force of equal magnitude is applied to a second object, then the object with more mass will accelerate less

• mass is intrinsic property of an object that does not depend on its location

  • remains the same whether the object is on earth, moon, outer space

The Force Due to Gravity: Weight

• if object’s weight is the ONLY force actin on an object, the object is said to be in free-fall

• Newton’s second law defines the weight, w

  • w = mg

• weight of an object is proportional to its mass

• g = 9.81 m/s².

• an object weighs slightly less at very high altitudes than it does at sea level

• g varies slightly with latitude because earth is not exactly spherical, but is slightly flattened at polls

• weight, unlike mass, is NOT an intrinsic property

• if there is no force to balance your weight, the force it must exert to balance your weight, your apparent weight, is zero

  • weightlessness is experienced by astronauts in orbiting satellites

Units of Force and Mass

kilogram is fundamental unit

unit of force: the newton

1N = (1 kg) (1 m/s²) = 1kg*m/s²

unified mass unit (u) = atomic mass (carbon-12)

1u = 1.660 540 × 10^-27 kg

9.81 N = 2.2 lbs

g = 32.2 ft/s²

Forces in Nature

• full power of newton’s 2nd law when combined with force laws that describe objects’ interactions

Fundamental Forces

1. Gravitational Force: the force of mutual attraction between objects

2. Electromagnetic Force: the force between electric charges

3. Strong Nuclear Force: the force between subatomic particles

4. Weak Nuclear Force: the force between subatomic particles during certain radioactive decay processes

• everyday forces are electromagnetic and gravitational

Action at a Distance

• fundamental forces of gravity and electromagnetism act between particles separated in space

  • action at a distance

• concept of a field acts as intermediary agent

Contact Forces

• many forces are exerted by objects indirect contact

  • electromagnetic in origin

    • exerted between surface molecules of objects in contact

Solids

• if surface is pushed against, it pushes back

• force perpendicular to contacting surfaces is called “normal force”

  • normal means perpendicular

• frictional force acts parallel to contacting surfaces

Springs

F_x = -k ∆x

• ∆x is amount spring is compressed or extended

• k is force constant

  • measure of stiffness of spring

• force exerts back is in opposite direction

• Hooke’s Law: an object at rest under the influences of forces that balance is said to be in static equilibrium

  • if small displacement results in a net restoring force towards equilibrium position, equilibrium is called stable equilibrium

  • nearly all small displacements obey Hooke’s Law

• molecular force of attraction between atoms in.a molecule or solid varies approximately linearly with the change in separation for small changes

• can use. two masses on a spring to model diatomic molecule or set of masses connected by springs to model a solid

Exercise in Dimensional Analysis

• an object of mass m oscillates at the end of an ideal spring of force constant k

• time for one complete oscillation is the period T

• assuming T depends on m and k, dimensional analysis finds form of the relationship

  • T = f(k,m)

T = 2πC√m/k

• C is dimensionless constant

Strings

• strings are used to pull things

• string is spring with large enough force constant that extension of string is negligible

• strings cannot push things

• magnitude of force that one segment of string exerts on adjacent segment is called tension

• magnitude of force equals tension

Constraints

• railroad car moving along track or sled on snow motion is called constraints

Problem Solving: Free-Body Diagrams

• connected objects are one

• free body diagram is diagram that shows all forces acting on object is drawn without its surroundings

• first need to determine direction of acceleration vector using kinematics

• a_x = F/m

  • from x component of Newton’s second law

• F_n = w

  • from y component of Newton’s second law

Newton’s Third Law

• when two objects interact, they evert forces on each other

  • 3rd law states that these forces are equal in magnitude and opposite in direction

• force always occurs in pairs

  • one force in pair is action and other is reaction

    • either can be called either

• if external force acting on object as action force, corresponding reaction force must act on different object

• no two external forces acting on single object can ever constitute an action-reaction pair

• weight due to attraction of earth

• table in between object and earth would have upward force on object that has downward attraction to earth to avoid collisions

  • action-reaction paired force between table and object as well

Problems With Two or More Objects

• in some problems two or more objects are in contact by a string or spring

  • solve problems by drawing free body diagrams for each object, then applying newton’s second law to each object

    • resultant equations together with any equations describing interactions and constraints are solved simultaneously for unknown quantities

• if objects are in direct contact, the forces they exert on each other must be equal and opposite

• two objects moving in straight line connected by string, acceleration components parallel to string are same for both objects

  • for each object, its motion parallel to the string is identical with that of the other object

  • if string passes over pulley or peg, “parallel to the string” means parallel with that segment of the string attached to the object

• tension is same throughout the entire length of the rope

• if taut rope of negligible mass changes direction by passing over a frictionless surface, the tension is the same throughout the rope

Friction

Static Friction

• when apply small horizontal force to large box resting, box may not move bc of force of static friction exerted by the floor on the box balancing force being applied

• f_s.max is proportional to the normal force exerted by one surface on the other

f_{s. max} = μ_sF_n

• μ_s is coefficient of static friction

  • depends on nature of surfaces in contact

• if horizontal force smaller than f_s.max on box, frictional force will just balance horizontal force

f_s ≤ μ_sF_n

Kinetic Friction

• if you push box hard enough, it will slide across the floor

• as box slides, floor exerts force of kinetic friction

  • also called sliding friction

• kinetic friction opposes motion

• to keep box sliding with constant velocity, must exert force that is equal in magnitude and opposite in direction to force of kinetic friction exerted by floor

• coefficient of kinetic friction is μ_k

  • ration of magnitudes of kinetic frictional force f_k and normal force F_n

f_k = μ_kF_n

• μ depends on nature of surfaces in contact

• experimentally, u_k is found to be less than μ_s and is approximately constant for speeds ranging from 1 cm/s to several meters per second

Rolling Friction

• when ideal, rigid wheels roll at constant speed along ideal, rigid horizontal road without slipping, no frictional force slows motion

• real road exerts force of rolling friction that opposes motion

• to keep wheel rolling with constant velocity, must exert force on wheel that is equal in magnitude and opposite in direction to the force of rolling friction exerted by the road

• coefficent of rolling friction is μ_r

  • ratio of magnitudes of rolling frictional force f_r and normal force F_n

f_r = μ_rF_n

• μ_r depends on nature of surfaces in contact and composition of wheel and road

  • typically 0.01 - 0.02 for rubber tires on concrete and 0.001 to 0.002 for steel wheels on steel rails

Friction Explained

• arises from the attraction of molecules between two surfaces that are in close contact

• nature of contact is electromagnetic

  • same as molecular bonding that holds and object together

• attractive force is negligible at distances of only a few atomic diameters

• when surfaces come into contact, they touch only at widely spaced prominences called asperities

• normal force exert4d by a surface is exerted at tips of asperities where force per unit area is very large

  • large enough to flatter tips of asperities

• under wide range of conditions, the microscopic area of contact is proportional to microscopic contact area, so it is also proportional to the normal force

Motion Along a Curved Path

• particle moving with speed v along curved path with radius of curvature r has acceleration component a_c = v²/r in centripetal direction (toward center of curvature) and an acceleration component a_t = dv/dt in the tangential direction

• net force is in direction of the acceleration

• component of net force in centripetal direction called centripetal force

  • name for net-force component perpendicular to direction of motion

Banked Curves

• if curved road is not horizontal but banked, normal force of the road will have component directed inward toward the center of the circle that will contribute to centripetal force

• banking angle can be chosen so that, for given speed, no friction is needed for a car to complete the curve

Drag Forces

• when object moves through a fluid such as air or water, fluid exerts a drag force (or retarding force) that opposes the motion of the object

• drag force depends on shape of object, properties of fluid, and speed of object relative to the fluid

• drag force increases as speed of object increases

• at low speeds drag force is approximately proportional to the speed of the object; at higher speeds, it is more nearly proportional to the square of the speed

• bvⁿ

  • b and n are constants

• an object dropped from rest and falling under gravitational force, drag force of magnitude bvⁿ

  • constant downward force mg and an upward force bvⁿ

  • at t = 0, speed is zero, drag force is zero and acceleration is g downward

  • as speed increases, drag force increases and acceleration becomes less than g

  • eventually, speed is great enough for magnitude of drag force bvⁿ to approach force of gravity mg

    • as this happens acceleration approaches zero and the speed approaches the terminal speed v_t

      • at terminal sped, drag force a=balances the weight force and acceleration is zero

• bv_tⁿ = mg

• solving for terminal speed

  • v_t = (mg/b)^1/n

    • the larger b, the smaller the terminal speed

• parachute is designed to maximize b so that the terminal speed will be small

• cars are designed to minimize b to reduce effect of wind resistance

• reimann’s sum of F_y = mg-bvⁿ = ma_y