ChemPhys 173/273

Unit 8: Newton's Laws Applications

Problem Set E

 

Overview:

Problem Set E targets your ability to analyze the motion of objects along inclined planes and to analyze the motion of two-body systems. The first 12 problems of the set involve inclined plane analysis; these problems are very difficult problems, all having their unique features which make them different than the others. The last 8 problems are two-body problems; they are relatively simple problems which provide an introduction to one of the most difficult topics of the unit.

 

Inclined Plane Problems

Problems #1 - #12 will target your ability to analyze objects positioned on inclined planes, either accelerating along the incline or in equilibrium. As in all problems in this set, the analysis begins with the construction of a free-body diagram in which forces acting upon the object are drawn. This is shown below on the left. Note that the force of friction is directed parallel to the incline, the normal force is directed perpendicular to the incline, and the gravity force is neither parallel nor perpendicular to the incline. It is common practice in Fnet = m•a problems to analyze the forces acting upon an object in terms of those which are along the same axis of the acceleration and those which are perpendicular to it. On horizontal surfaces, we would look at all horizontal forces separate from those which are vertical. But on inclined surfaces, we would analyze the forces parallel to the incline (along the axis of acceleration) separate from those which are perpendicular to the incline. Since the force of gravity is neither parallel nor perpendicular to the inclined plane, it is imperative that it be resolved into two components of force which are directed parallel and perpendicular to the incline. This is shown on the diagram below in the middle. The formulas for determining the components of the gravity force parallel and perpendicular to the inclined plane are listed on the diagram. Once the components are found, the gravity force can be ignored since it has been substituted for by its components; this is illustrated in the diagram below on the right.

 

Once the gravity force has been resolved into its perpendicular and parallel components, the problem is approached like any Fnet = m•a problem. The net force is determined by adding all the forces as vectors. The forces directed perpendicular to the incline balance each other and add to zero. For the more common cases in which there are only two forces perpendicular to the incline, one might write this as:

Fnorm = Fperpendicular

The net force is therefore the result of the forces directed parallel to the incline. As always, the net force is found by adding the forces in the direction of acceleration and subtracting the forces directed opposite of the acceleration. In the specific case shown above for an object sliding down an incline in the presence of friction,

Fnet = Fparallel - Ffrict

Once the net force is determined, the acceleration can be calculated from the ratio of net force to mass.

There are a variety of situations that could occur for the motion of objects along inclined planes. There are situations in which a force is applied to the object upward and parallel to the incline to either hold the object at rest or to accelerate it upward along the incline. Whenever there is a motion up the inclined plane, friction would oppose that motion and be directed down the incline. The net force is still determined by adding the forces in the direction of acceleration and subtracting the forces directed opposite of the acceleration. In this case, the net force is given by the following equation:

Fnet = Fapp - Fparallel - Ffrict

Specific discussions of each of the myriad of possibilities is not as useful as one might think. Most often such discussions cause physics students to focus on the specifics and to subsequently miss the big ideas which underlay every analysis regardless of the specific situation. Every problem can be (and should be) approached in the same manner: by drawing the free-body diagram showing all the forces acting upon the object, to resolve the gravity force into components parallel and perpendicular to the incline and to write the Fnet expression by adding the forces in the direction of acceleration and subtracting the forces directed opposite of the acceleration.

 

Two-Body Problems

Problems #13 - #20 will target your ability to analyze the motion of two-body systems. Two-body problems (and three-body problems) are typically approached by completing two distinct analyses. There is a system analysis in which the two objects are considered to be a single object moving (or accelerating) together as a whole. The mass of the system is the sum of the mass of the individual objects. If an acceleration is involved, the acceleration of the system is the same as that of the individual objects. In analyzing the forces acting upon the system, disregard any forces which act between the objects themselves since they are part of the system.

In addition to the system analysis, there is an individual object analysis in which an object is isolated and considered as a separate entity. Free-body diagrams are constructed in the usual manner and individual forces acting upon the object are identified and calculated. In general, a system analysis is usually performed to determine the acceleration of the system (and therefore of the individual objects). And a individual object analysis is performed to determine the value of any force which acts between the two objects - for example, contact forces or tension forces.

 

Additional Readings/Study Aids:

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.

Inclined Planes | Two-Body Problems (coming soon)

Forces | Mass and Weight | Drawing Free Body Diagrams | Kinematic Equations

 

View Sample Problem Set.

 

Problem

Description

Audio Link
1

Determination of the applied force required to pull a furniture crate up an incline at constant speed; friction is involved.

2

Determination of the applied force required to pull a shopping cart up a frictionless incline with a stated acceleration.

3

Determine the coefficient of friction for an incline if a box slides down it with a specified acceleration.

4

Determination of the applied force required to pull a box up an incline at constant speed; friction is involved.

5

Determine the mass of a box if a specified force is required to prevent it from sliding down a frictionless incline.

6

Combine Newton's laws and kinematics to determine the final speed of a wagon pulled up a hill by a tension force; neglect friction.

7

Calculate and compare the acceleration of two boxes of different mass sliding down the same frictionless incline.

8

Calculate the acceleration of a car rolling down an incline plane; neglect resistance forces.

9

Referring to the previous problem; use kinematics to determine the time to roll to the bottom of the incline.

10

Difficult analysis in which an applied force is exerted on a sled at an angle to the incline to pull it along the surface at a constant velocity; friction is involved and the applied force value must be calculated.

11

Determine the coefficient of static friction from knowledge of the maximum angle of incline allowed before an eraser begins to slide from rest down an incline.

12

Referring to the previous problem; determine the coefficient of sliding friction from knowledge of the angle at which an eraser slides down an incline at constant speed.

13

Analysis of a two-body system to determine the acceleration along a friction-free surface.

14

Referring to the previous problem; determine the tension in a string which connects the two objects of a two-body system.

15

Analysis of a two-body system to determine the acceleration along a friction-free surface.

16

Referring to the previous problem; determine the tension in a cord which connects the two objects of a two-body system.

17

Analysis of a two-body system to determine the acceleration along a rough surface having friction.

18

Referring to the previous problem; determine the tension in a string which connects the two objects of a two-body system.

19

Analysis of the constant velocity motion of a two-body system to determine the tension force in the rope which connects the two bodies.

20

Referring to the previous problem; determination of the force of friction on the back object of the two-body system.

 

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