ChemPhys 173/273

Unit 8: Newton's Laws Applications

Problem Set F

Overview:

Problem Set F targets your ability to use Newton's second law to analyze a wealth of physical situations. The first 15 problems of the set involve two-body analyses. The remaining problems serve as an excellent review of the many concepts present in this unit on Newton's laws applications: addition of forces to determine the resultant, equilibrium analysis, motion along inclined planes, and horizontal accelerations caused by forces at angles to the horizontal. Even some of the two-body problems involve physical situations which provide a review of mathematical principles from previous problem sets; some involve inclined planes or the application of forces at angles to the horizontal.

Solving Two-Body Problems

Two-body problems are problems pertaining to the motion of two objects which are connected together or somehow interacting in such a manner as to move together as a single unit. The problems could pertain to two blocks connected by a wire (string, cable, etc.) or simply two blocks touching each other as they are slid across a table top. Because the two objects are somehow connected, they individually accelerate at the same rate. Subsequently knowing the acceleration of one of the objects provides the acceleration of the second object.

There are two basic approaches to solving a two-body problem. One approach involves conducting a system analysis in which you consider the two bodies to simply be a single object. In your mind (and on paper), a box is drawn around the system defined as the two objects and any wire, string or cable which connects them.

A free-body diagram for the system is then drawn and only forces acting from outside the system are considered as effecting the acceleration of the system. If there is a string connecting the two objects, the tension force from the string is not considered since is it part of the system and as such cannot exert an external force on the system. In a system analysis, Newton's second law is applied by stating that the sum of all the forces from outside the system is equal to the total mass of the system (i.e., the combined mass of both objects) multiplied by the acceleration.

A second approach to solving a two-body problem involves conducting an individual body analysis. A single object is analyzed individually to determine the forces acting upon that object. A free-body diagram is drawn and Newton's second law is applied in the usual manner. While a contact force between the two objects or a tension force resulting from a string connecting the two objects are not considered in a system analysis, they must be considered in an individual body analysis. As is the usual case for a free-body analysis, one can apply Newton's laws for an individual object by stating that the net force is the sum of the forces in the direction of the acceleration minus the values of all forces which oppose the acceleration; this net force is equal to the mass of the object under consideration multiplied by its acceleration.

A two-body problem typically involves two unknown quantities. Thus there is a need for two equations in order to solve for those two unknowns. To develop two equations to solve for the two unknowns, one could either:

1. conduct two individual-body analyses and apply Newton's second law to each of the two individual objects, thus generating two equations for solving for the two unknowns; or
2. conduct a system analysis for the two objects considered as a single unit and one individual body analysis for either one of the objects, thus generating two equations for solving for the two unknowns.

Solving the Other Problems Not Involving Two Bodies (Problems #16-20)

The last five problems of Set F are simply review problems which encompass most of the concepts in the Newton's laws unit. Problems #16 And #17 pertain to forces at angles to cause an acceleration; kinematic equations must be blended with Newton's laws to determine a kinematic quantity. The Set D overview page would be useful reading to review these ideas. Problem #18 involves a terminal velocity situation in which an individual force value must be calculated; the Set B overview page would be useful reading to review these ideas. Problem #19 involves the acceleration of an object along an inclined plane; a Newton's second law analysis must be combined with kinematic equations to determine a kinematic quantity. The Set E overview page would be useful reading to review these ideas. Problem #20 is a straight-forward 1-dimensional application of Newton's second law with a two-body slant and the use of a kinematic equation. The Set B overview page would be useful reading to review these ideas.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of mathematics of inclined planes and two-body problems.

Two-Body Problems (coming soon)

View Sample Problem Set.

 Problem Description Audio Link 1 Determine the tension in the rope which vertically accelerates a two-body system; the rope acts from outside the system. 2 Referring to the previous problem; determine the tension in the rope which holds the two bodies of the system together. 3 Determination of the tension in a tow truck-car system which is accelerating horizontally. 4 Determination of the acceleration of a two-body system consisting of a mass on a frictionless surface connected to a second mass by a string which extends over a pulley. 5 Referring to the previous problem; determination of the tension in the string. 6 Determination of the tension in a string which connects two masses on opposite sides of a pulley (a class Atwood's machine). 7 Referring to the previous problem; determination of the two-body system. 8 Referring to the previous problem; application of kinematics to determine the distance that the hanging mass will fall in a given time. 9 Determination of the acceleration of a two-body system consisting of a mass on a rough surface connected to a second mass by a string which extends over a pulley. 10 A complex two-body analysis of a car on an inclined plane connected to a counterweight by a cable which extends over a pulley; the acceleration must be determined. 11 Referring to the previous problem; determination of the tension in the cable connecting the car and the counterweight. 12 Referring to the previous problem; determination of the counterweight necessary to allow a specified acceleration down the inclined plane. 13 Complex analysis which combines kinematic information with a force analysis of a two-body system involving an incline, a pulley and a connecting cable; must determine the mass of one of the objects. 14 A complex two-body analysis to determine the tension in an angled rope which pulls on the two-body system to keep one of the bodies from accelerating across a rough surface despite the pull of a second body which hangs over a pulley. 15 Complex analysis of a two-body system which vertically accelerates at a stated rate; must determine the force which holds the two objects together. 16 Complex analysis of three forces acting upon an object to determine the resultant force and acceleration and ultimately the speed after a specified time. 17 Combine a force analysis and a kinematic analysis to determine the final speed of an object which is acted upon by an angled force to accelerate horizontally across a frictionless surface. 18 Determination of the air resistance force on a air boarder if given the mass and acceleration. 19 Combine a force analysis and a kinematic analysis to determine the final speed of a skier along a frictionless surface. 20 Combine a force analysis and a kinematic analysis to determine the final speed of a car acted upon by two horizontal forces for a given amount of time.

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