# Chapter 3uniform Circular Motionmr.'s Learning Website

### Learning Objectives

- Chapter 3 Uniform Circular Motionmr.'s Learning Websites
- Chapter 3 Uniform Circular Motionmr.'s Learning Website Examples
- Chapter 3 Uniform Circular Motionmr.'s Learning Website Template

By the end of this section, you will be able to:

- Learn uniform circular motion physics with free interactive flashcards. Choose from 500 different sets of uniform circular motion physics flashcards on Quizlet.
- Visit for chapter 8, class 9 motion notes. In physics motion is change in position of object over time. Motion is described by terms like distance, displacement velocity, acceleration etc.

^ For the above Powerpoint, the negative and positive conventions are wrong, make sure you do the question correctly. This chapter deals with the simplest form of curved motion, uniform circular motion, motion in a circular path at constant speed.Studying this topic illustrates most concepts associated with rotational motion and leads to the study of many new topics we group under the name rotation. On this page I put together a collection of circular motion problems to help you understand circular motion better. The required equations and background reading to solve these problems is given on the rotational motion page. Refer to the figure below for problems 1-6.

- Compare simple harmonic motion with uniform circular motion.

Figure 1. The horses on this merry-go-round exhibit uniform circular motion. (credit: Wonderlane, Flickr)

Figure 2. The shadow of a ball rotating at constant angular velocity ω on a turntable goes back and forth in precise simple harmonic motion.

There is an easy way to produce simple harmonic motion by using uniform circular motion. Figure 2 shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions (*ω* constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in Figure 2, is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.

Figure 3 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity *ω*. The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position *x* and moves to the left with velocity *v*. The velocity of the point P around the circle equals [latex]bar{v}_{text{max}}[/latex].The projection of [latex]bar{v}_{text{max}}[/latex] on the *x*-axis is the velocity *v* of the simple harmonic motion along the *x*-axis.

In Figure 3 we see that a point P moving on a circular path with a constant angular velocity *ω* is undergoing uniform circular motion. Its projection on the x-axis undergoes simple harmonic motion. Also shown is the velocity of this point around the circle,[latex]bar{v}_{text{max}}[/latex], and its projection, which is *v*. Note that these velocities form a similar triangle to the displacement triangle.

To see that the projection undergoes simple harmonic motion, note that its position *x* is given by

*x* = *X* cos *θ, *

where *θ* = *ωt*, *ω* is the constant angular velocity, and *X* is the radius of the circular path. Thus,

*x* = *X* cos *ωt.*

The angular velocity *ω* is in radians per unit time; in this case 2π radians is the time for one revolution *T*. That is, [latex]omega=frac{2pi}{T}[/latex]. Substituting this expression for *ω*, we see that the position *x* is given by:

[latex]x(t)=cosleft(frac{2pi{t}}{T}right)[/latex].

Figure 4. The position of the projection of uniform circular motion performs simple harmonic motion, as this wavelike graph of *x* versus *t* indicates.

This expression is the same one we had for the position of a simple harmonic oscillator in Simple Harmonic Motion: A Special Periodic Motion. If we make a graph of position versus time as in Figure 4, we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the *x*-axis.

Now let us use Figure 3 to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements (*X*, *x*, and [latex]sqrt{X^2-x^2}[/latex]) are similar right triangles. Taking ratios of similar sides, we see that

[latex]displaystylefrac{v}{v_{text{max}}}=sqrt{frac{X^2-x^2}{X}}=sqrt{1-frac{x^2}{X^2}}[/latex]

We can solve this equation for the speed *v* or

[latex]displaystyle{v}={v_{text{max}}}sqrt{1-frac{x^2}{X^2}}[/latex].

This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Energy and the Simple Harmonic Oscillator.You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.

Finally, let us consider the period *T* of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle 2π*X* divided by the velocity around the circle, *v*_{max}. Thus, the period *T* is

[latex]T=frac{2pi{X}}{v_{text{max}}}[/latex].

We know from conservation of energy considerations that

Engineering manual pdf. [latex]v_{text{max}}=sqrt{frac{k}{m}}X[/latex].

Solving this equation for

[latex]frac{X}{v_{text{max}}}[/latex] gives [latex]frac{X}{v_{text{max}}}=sqrt{frac{m}{k}}[/latex].

Substituting this expression into the equation for *T* yields

[latex]T=2pisqrt{frac{m}{k}}[/latex].

### Chapter 3 Uniform Circular Motionmr.'s Learning Websites

Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.

Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

### Check Your Understanding

Identify an object that undergoes uniform circular motion. Describe how you could trace the simple harmonic motion of this object as a wave.

#### Solution

A record player undergoes uniform circular motion. You could attach dowel rod to one point on the outside edge of the turntable and attach a pen to the other end of the dowel. As the record player turns, the pen will move. You can drag a long piece of paper under the pen, capturing its motion as a wave.

## Section Summary

A projection of uniform circular motion undergoes simple harmonic oscillation.

### Problems & Exercises

- (a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of 1.50 × 10
^{6}N/m, if the amplitude of the bounce is 0.200 cm? (b)What is the maximum energy stored in the spring? - A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?
- At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of [latex]frac{x}{X}[/latex] give [latex]v=pmfrac{{v}_{text{max}}}{2}[/latex], where
*X*is the amplitude of the motion? - A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?

### Selected Solutions to Problems & Exercises

1. (a) 0.266 m/s; (b) 3.00 J

3. [latex]pmfrac{sqrt{3}}{2}[/latex]

### Chapter 3 Uniform Circular Motionmr.'s Learning Website Examples

Many motions, such as the arc of a bird’s flight or Earth’s path around the Sun, are curved. Recall that Newton’s first law tells us that motion is along a straight line at constant speed unless there is a net external force. We will therefore study not only motion along curves, but also the forces that cause it, including gravitational forces. In some ways, this chapter is a continuation of Dynamics: Newton’s Laws of Motion as we study more applications of Newton’s laws of motion.

Figure 1. This Australian Grand Prix Formula 1 race car moves in a circular path as it makes the turn. Its wheels also spin rapidly—the latter completing many revolutions, the former only part of one (a circular arc). The same physical principles are involved in each. (credit: Richard Munckton)

### Chapter 3 Uniform Circular Motionmr.'s Learning Website Template

This chapter deals with the simplest form of curved motion, *uniform circular motion*, motion in a circular path at constant speed. Studying this topic illustrates most concepts associated with rotational motion and leads to the study of many new topics we group under the name *rotation*. Pure *rotational motion* occurs when points in an object move in circular paths centered on one point. Pure *translational motion* is motion with no rotation. Some motion combines both types, such as a rotating hockey puck moving along ice.