PHY 151: Lecture Mass 5.4 Newton’s Second Law 5.5 Gravitational Force and Weight 5.6 Newton’s Third Law

PHY 151: Lecture 5 The Laws of Motion 5.3 Mass

Inertia and Mass - 1 The tendency of an object to resist any attempt to change its velocity is called inertia Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity Masses can be defined in terms of the accelerations produced by a given force acting on them: –The magnitude of the acceleration acting on an object is inversely proportional to its mass

Mass - 2 Mass is an inherent property of an object Mass is independent of the object’s surroundings Mass is independent of the method used to measure it Mass is a scalar quantity –Obeys the rules of ordinary arithmetic The SI unit of mass is kg

Mass vs. Weight Mass and weight are two different quantities Weight is equal to the magnitude of the gravitational force exerted on the object –Weight will vary with location Example: –w earth = 180 lb; w moon ~ 30 lb –m earth = 2 kg; m moon = 2 kg

PHY 151: Lecture 5 The Laws of Motion 5.4 Newton’s Second Law

Newton’s Second Law Newton’s Statement of the Law –A measure of motion is momentum = p = mv –The greater the “motion” the greater is mv –Rate of change of momentum =  F –  F =  p/  t =  (mv)/  t = [(mv) f – (mv) i ]/  t –When viewed from an inertial reference frame –Force, momentum and velocity are vectors –  F means sum of external forces

Newton’s Second Law Force - Acceleration Contemporary Statement of the Law –Mass does not change, then –  F =  p/  t =  (mv)/  t = m  v/  t = ma –  F = ma

Newton’s Second Law Definitions External Forces –Net force includes only the forces that the environment exerts on the object Internal Forces –Forces that one part of an object exerts on another part of the object –These are not included in the second law

More About Newton’s Second Law is the net force –This is the vector sum of all the forces acting on the object May also be called the total force, resultant force, or the unbalanced force Newton’s Second Law can be expressed in terms of components: –  F x = m a x –  F y = m a y –  F z = m a z Remember that ma is not a force –The sum of the forces is equated to this product of the mass of the object and its acceleration

Unit of Force SI unit of force is the Newton (N) This a derived unit = kg(m/s 2 ) Weight is the force of gravity If you held a weight of 3.5 ounces in the palm of your hand you would feel the force of 1 Newton This is very little force

Newton’s Second Law Example 1 A 3.0-N net force is applied to 1.5-kg mass What is the object’s acceleration? –Magnitude F = ma a = F/m = 3.0 N / 1.5 kg = 2.0 m/s 2 –Direction In the direction of the net force

Newton’s Second Law Example 2 Engine of a 1.0-kg toy plane exerts a 15-N forward force Air exerts a 8.0-N resistive force on the plane The resistive force of the air is negative (-) What is the magnitude of the acceleration of the plane?  F = ma  15 – 8.0 = 1.0a  7.0 = 1.0a  a = 7.0/1.0 = 7.0 m/s 2

Newton’s Second Law Example 3 A boy pulls a box of mass 30 kg with a force of 25 N that makes an angle of 30 0 with the horizontal direction What is the acceleration of the box?  We are only interested in x-component (horizontal) of force and acceleration  F = ma  25cos(30) = 30a  a = 25cos(30)/30 = 0.72 m/s 2

Newton’s Second Law Example 4 Two boats pull a 75.0-kg water skier Each boat pulls with a force of 600 N One boat pulls 45 0 above the horizontal axis The other boat pull 45 0 below the horizontal axis The skier travels at a constant velocity What is the magnitude of the retarding force between the water and the skis?  Let direction of motion be in the +x direction  Work with vector x-components  Ncos(45) + Ncos(45) - F retarding = ma = 0  2(600)cos(45) = F retarding = 849 N

PHY 151: Lecture 5 The Laws of Motion 5.5 Gravitational Force and Weight

Gravitational Force The gravitational force,, is the force that the earth exerts on an object This force is directed toward the center of the earth From Newton’s Second Law: – Its magnitude is called the weight of the object –Weight = F g = mg

More About Weight Because it is dependent on g, the weight varies with location g, and therefore the weight, is less at higher altitudes –This can be extended to other planets, but the value of g varies from planet to planet, so the object’s weight will vary from planet to planet Weight is not an inherent property of the object –The weight is a property of a system of items: the object and the Earth Note about units: –Kilogram is not a unit of weight –1 kg = 2.2 lb is an equivalence valid only on the Earth’s surface

Gravitational Mass vs. Inertial Mass In Newton’s Laws, the mass is the inertial mass and measures the resistance to a change in the object’s motion In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth Experiments show that gravitational mass and inertial mass have the same value

PHY 151: Lecture 5 The Laws of Motion 5.6 Newton’s Third Law

Newton’s Statement of the Law –A system consists of objects 1 and 2 –Objects 1 and 2 exert internal forces on each other –There is no external force –F 12 is the internal force of object 1 on object 2 –Apply Newton’s Second Law to the system   (p 1 + p 2 )/  t = 0 Newton’s Third Law - 1

Newton’s Statement of the Second Law   (p 1 + p 2 )/  t = 0   p 1 /  t = -  p 2 /  t = 0  Apply Newton’s Second Law separately to objects 1 and 2  F 21 = - F 12  Forces are equal in magnitude  Forces are opposite in direction  Sum of momentums of two objects is conserved (unchanged) Newton’s Third Law - 2

Newton’s Third Law - 3 Contemporary Statement of the Law  When one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body

Newton’s Third Law – 4 The action force is equal in magnitude to the reaction force and opposite in direction –One of the forces is the action force, the other is the reaction force –It doesn’t matter which is considered the action and which the reaction –The action and reaction forces must act on different objects and be of the same type

Action-Reaction Examples - 1 The force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to exerted by object 2 on object 1. 

Action-Reaction Examples - 2 The normal force (table on monitor) is the reaction of the force the monitor exerts on the table –Normal means perpendicular, in this case The action (Earth on monitor) force is equal in magnitude and opposite in direction to the reaction force, the force the monitor exerts on the Earth

Forces on the Object In a free body diagram, you want the forces acting on a particular object –Model the object as a particle The normal force and the force of gravity are the forces that act on the monitor

Free Body Diagram The most important step in solving problems involving Newton’s Laws is to draw the free body diagram Be sure to include only the forces acting on the object of interest Include any field forces acting on the object Do not assume the normal force equals the weight

Free Body Diagrams and Particle Model The particle model is used by representing the object as a dot in the free body diagram The forces that act on the object are shown as being applied to the dot The free body helps isolate only those forces acting on the object and eliminate the other forces from the analysis