ChemPhys 173/273

Unit 9: Circular Motion and Gravitation

Problem Set A

Overview:

Problem Set A targets your ability to use vector principles and Newton's laws of motion in order to analyze the motion of objects moving in circles or along curved paths. Perhaps more than any other topic, the motion of objects in circles represents a strong blend of conceptual ideas with mathematical relationships. It would be difficult to be highly successful on most topics in Physics without a strong mathematical base; that is certainly true of this topic. But it is just as true to say that it would very difficult to be highly successful with mathematical problems regarding motion in circles without a strong conceptual base. Most of this problem set involves an application of Newton's laws to the motion of objects along circular paths. There are three basic ideas and skills which you will need to master to be successful on this set of problems. They are:

• Motion Characteristics for Objects Moving in Circles

Objects moving in circles have a speed which is equal to the distance traveled per time of travel. The distance around a circle is equivalent to a circumference and calculated as 2•pi•R where R is the radius. The time for one revolution around the circle is referred to as the period and denoted by the symbol T. Thus the average speed of an object in circular motion is given by the expression 2•pi•R / T. Often times the problem statement provides the rotational frequency in revolutions per minute or revolutions per second. Each revolution around the circle is equivalent to a circumference of distance. Thus, multiplying the rotational frequency by the circumference allows one to determine the average speed of the object.

The acceleration of objects moving in circles is based primarily upon a direction change. The actual acceleration rate is dependent upon how rapidly the direction is being change and is directly related to the speed and inversely related to the radius of the turn. It ends up that the acceleration is given by the expression v2 / R where v is the speed and R is the radius of the circle.

The equations for average speed (v) and average acceleration (a) are summarized below.

 v = d / t = 2•pi•R / T = frequency • 2•pi•R a = v2 / R

• Directional Quantities for Objects Moving in Circles

A successful mathematical analysis of objects moving in circles is heavily dependent upon a conceptual understanding of the direction of the acceleration and net force vectors. Movement along a circular path requires a net force directed towards the center of the circle. At every point along the path, the net force must be directed inwards. While there may be an individual force pointing outward, there must be an inward force which overwhelms it in magnitude and meets the requirement for an inward net force. Since net force and acceleration are always in the same direction, the acceleration of objects moving in circles must also be directed inward.

• Free Body Diagrams and Newton's Second Law

The last half of this problem set includes several problems in which a force analysis is conducted upon an object moving in circular motion. The goal of the analysis is either to determine the magnitude of an individual force acting upon the object or to use the values of individual forces to determine an acceleration. Like any force analysis problem, these problems should begin with the construction of a free-body diagram showing the type and direction of all forces acting upon the object. From the diagram, an Fnet = m•a equation can be written. When writing the equation, recall that the Fnet is the vector sum of all the individual forces. It is best written by adding all forces acting in the direction of the acceleration (inwards) and subtracting those which oppose it. Two examples are shown in the graphic below.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Forces | Mass and Weight | Drawing Free-Body Diagrams

View Sample Problem Set.

 Problem Description Audio Link 1 Routine calculation of the speed of an object moving in circular motion; diameter and period are given. 2 Calculation of the acceleration of an object moving in circular motion; speed and diameter are given; requires some attention to units. 3 Determination of the acceleration of an object moving in a circle; information about diameter and the number of revolutions in a specific time are given. 4 Determination of the acceleration of an object moving in a circle; the radius and rotational frequency are given. 5 Determination of the linear speed of an object moving in a circle; the radius and rotational frequency are given. 6 Referring to the previous problem; determine the acceleration. 7 Determination of the acceleration of an object moving in a circle; the radius and rotational frequency are given. 8 Referring to the previous problem; determine the acceleration at a different rotational frequency 9 Determination of the rotational frequency of an object moving in a circle; the diameter and the acceleration are given. 10 Determination of the speed of an object moving in a circle; the force, mass and radius are given. 11 Determination of the speed of an object moving in a horizontal circle; requires a force analysis to determine acceleration. 12 Determination of the speed of an object at one location while moving in a vertical circle; requires a force analysis to determine acceleration. 13 Determination of the radius of a circle through which an object is moving; acceleration and speed are given. 14 Referring to the previous problem; determination of the net force from mass and acceleration. 15 Determination of the net force by combining trigonometry and a force analysis. 16 Referring to the previous problem; determination of the acceleration from the net force and mass. 17 Determination of the normal force acting upon an object at the bottom of a vertical circle; requires a force analysis. 18 Referring to the previous problem; determine the maximum speed at the top of a vertical circle; requires a force analysis. 19 Determination of the minimum speed of an object at the top of a vertical circle; requires a force analysis. 20 Determination of the radius of curvature at the top of a roller coaster loop; requires a force analysis to determine the acceleration.

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