General Physics I Forces

Dynamics Isaac Newton (1643-1727) published Principia Mathematica in 1687. In this work, he proposed three laws of motion based on the concept of FORCE. A force is a push or a pull. A force has magnitude & direction (vector). Law 1: An object subject to no external forces is at rest or moves with a constant velocity.» If no forces act, there is no acceleration. Law 2: For any object, F NET = F = ma Law 3: Forces occur in pairs: F A,B = - F B,A (For every action there is an equal and opposite reaction.)

Newton s Second Law For any object, F NET = F = ma. The acceleration a of an object is proportional to the net force F NET acting on it. The constant of proportionality is called mass, denoted m. This is the definition of mass. The mass of an object is a constant property of that object, and is independent of external influences. Force has units of [kg m/s 2 ] = N (Newton)

Newton s Second Law Components of F = ma : F X = ma X F Y = ma Y Adding forces is like adding vectors a NET F 1 F 2 F NET = ma NET Suppose we know m and F X, we can solve for a X and apply the result to the things we learned about kinematics over the last few weeks: v x 2 x v 0x 1 x x0 v0xt axt 2 v v 2a ( x x 2 ox a x t x 2 o )

Example: Pushing a Box on Ice. A girl is pushing a heavy box (mass m) across a sheet of ice (horizontal & frictionless). She applies a force in the x direction. If the box starts at rest, what is its speed v after being pushed a distance d? v F m a d x

Example: Pushing a Box on Ice... Where do we start? v F m a d x

Forces We will consider two kinds of forces: Contact force: This is the most familiar kind. I push on the desk (Applied Force). The ground pushes on the chair (Normal Force). Resistance to Motion (Frictional Force) Action at a distance (Non-Contact Forces): Gravity (a little bit in this lecture) Electricity (next semester) Magnetism (next semester)

Contact Forces & Newton's Third Law: Objects in contact exert forces. Convention: F a,b means the force acting on a due to b. F m,w F w,m So F w,m means the force on the wall due to the man. F A, B = - F B, A is true for all forces F f,m F m,f

Example of Good Thinking Consider only the box as the system! F on box = ma box = F b,m Free Body Diagram (more on this soon ) a box F m,b F b,m ice

Newton s 3rd Law Two blocks are stacked on the ground. How many action-reaction pairs of forces are present in this system? a (a) 2 (b) 3 b (c) 4

The Free Body Diagram Newton s 2nd Law says that for an object F = ma. Key phrase here is for an object. So before we can apply F = ma to any given object we isolate the forces acting on this object. Consider the following case: What are the forces acting on the plank? P = Plank F = Floor W = Wall E = Earth

The Free Body Diagram... Isolate the plank from the rest of the world.

The Free Body Diagram... The forces acting on the plank should reveal themselves... F P,W From Newton s Second Law: F P,E = ma = mg F P,F F P,E Where: g 9.81 m / s 2

More... In this example the plank is not moving... It is certainly not accelerating! So F NET = ma becomes F NET = 0 F P,W F P,W + F P,F + F P,E = 0 F P,F F P,E This is the basic idea behind statics (F NET = 0 cases) We ll hit some of this here but will see more when we cover torque!

Free-Body Diagram & The Normal Indicate the forces acting on the block. Is the block moving? m The NORMAL is ALWAYS perpendicular to the surface doing the pushing No Surface = No Normal Multiple Surfaces = Multiple Normals

More Free-Body Diagram Examples ΣF x : ΣF y :

Normal Force A block of mass m rests on the floor of an elevator that is accelerating upward. What is the relationship between the force due to gravity and the normal force on the block? (a) N > mg (b) N = mg (c) N < mg a m

New Topic: Friction What does it do? It opposes relative motion! How do we characterize this in terms we have learned? Friction results in a force in the direction opposite to the direction of relative motion! Parallel to surface. Perpendicular to Normal force. y x

Surface Friction... Friction is caused by the microscopic interactions between the two surfaces:

Model for Sliding Friction The direction of the frictional force vector is perpendicular to the normal force vector N. The magnitude of the frictional force vector f F is proportional to the magnitude of the normal force N. f F = K N ( = K mg in the previous example) The heavier something is, the greater the friction will be...makes sense! The constant K is called the coefficient of kinetic friction. These relations are all useful APPROXIMATIONS to messy reality.

Model... Dynamics: x : Recall from Newton s 2 nd Law: F x m = a x y : so N y F ma x f K N mg

Static Friction... So far we have considered friction acting when the two surfaces move relative to each other- I.e. when they slide.. We also know that it acts in when they move together: the static case. In these cases, the force provided by friction will depend on the OTHER forces on the parts of the system. The maximum possible force that the friction between two objects can provide is f MAX = S N, where s is the coefficient of static friction. So f F S N. As one increases F, f F gets bigger until f F = S N and the object starts to move. N y F x f F mg *** Just like in the sliding case except a = 0 ***

Additional comments on Friction: Since f F = N, the force of friction does not depend on the area of the surfaces in contact. This is a surprisingly good rule of thumb, but not an exact relation By definition, it must be true that S K for any system (think about it...).

Springs Springs are objects that exhibit elastic behavior An ideal spring is: Massless = the mass of the spring is negligible. The applied force (F applied ) required to compress/stretch is proportional to the displacement of the spring from its unstrained length (x) or F applied = kx. Where k is called the spring constant (or stiffness of the spring) To stretch/compress a spring, the spring exerts a restoring force of equal & opposite magnitude (reaction force, F) against the stretching/compressing force or F s = -kx {this is referred to as Hooke s Law!}

The (Elastic) Restoring Force (& Newton s 3 rd Law) Action: Applied force is proportional to displacement of the spring: F applied = kx Reaction: Restoring force is equal/opposite to applied force: F s = -F applied = -kx

Tools: Ropes & Strings Can be used to pull from a distance. Tension (T) at a certain position in a rope is the magnitude of the force acting across a cross-section of the rope at that position. The force you would feel if you cut the rope and grabbed the ends. An action-reaction pair. T cut T T

Tools: Ropes & Strings... An ideal (massless) rope has constant tension along the rope. T T If a rope has mass, the tension can vary along the rope For example, a heavy rope hanging from the ceiling... T = T g T = 0 We will deal mostly with ideal massless ropes.

Tools: Ropes & Strings... The direction of the force provided by a rope is along the direction of the rope: a y = 0 (box not moving) m

Problem: Two strings & Two Masses on horizontal frictionless floor: Given T 1, m 1 and m 2, what are a and T 2? a x T 2 m 2 m 1 T 1

Problem: Tension and Angles --Statics-- A box is suspended from the ceiling by two ropes making an angle with the horizontal. What is the tension in each rope? 1 2 m Hint: How many dimensions are in play for this diagram?

1 2 m

Tools: Pegs & Pulleys Used to change the direction of forces An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: F 1 ideal peg or pulley F 1 = F 2 F 2

Horizontal Mass and a Hanging Mass Given M 1 and M 2, what are a and T? NOTE (New Rule): Direction of Motion = Positive Direction!!

The Inclined plane Consider a block of mass m that slides down a frictionless ramp that makes angle with respect to the horizontal having an acceleration a. Illustrate the free-body diagram in order to solve for a? Define convenient axes parallel and perpendicular to the plane: Ex: Acceleration a is in x direction only. a m

a m

Inclined Plane with Friction: m a

Two inclined planes - Dynamics Illustrate the Free-Body Diagram m 1 m 2 1 2

End of Forces Lecture