Recitation problems



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RECITATION PROBLEMS

CHAPTER 3

WEEK 4
3-3

A dragonfly flies from point A to point B along the path shown in the following figure in 1.50 s. (a) Find the x and y components of its position vector at point A. (b) What are the magnitude and direction of its position vector at A? (c) Find the x and y components of the dragonfly’s average velocity between A and B. (d) What are the magnitude and direction of its average velocity between these two points?



Coordinates for point A are (2.0 m, 1.0 m) and for point B they are (10.0 m, 6.0 m).

  1. At A, x = 2.0 m, y = 1.0 m.

  2. The magnitude of position vector is and

counterclockwise from the +x axis.



  1. The x component of the dragonfly’s average velocity between A and B:



  1. The magnitude of average velocity between A and B:


The direction of average velocity vector:

counterclockwise from the +x axis.
3-5

An athlete starts at point A and runs at a constant speed of 6.0 m/s around a round track 100 m in diameter, as shown in the following figure. Find the x and y components of this runner’s average velocity and average acceleration between points (a) A and B, (b) A and C, (c) C and D, and (d) A and A (full lap). (e) Calculate the magnitude of the runner’s average velocity between A and B. Is his average speed equal to the magnitude of his average velocity. Why or why not? (f) How can his velocity be changing if he is running at constant speed?



(a)

The time for one full lap is




A to B is one-quarter lap and takes


(b)

The time for A to C (half of the lap) is




X and y components for the average velocity and average acceleration are as follows:

(c)

The time for C to D is





(d)

For A to A (full lap):



so , and so
(e)

For A to B, .

Since the speed is constant, the average speed is 6.0 m/s. The distance traveled is larger than the displacement and hence the average speed is larger than the magnitude of the average velocity.
(f)

Here, the direction of the velocity vector is changing. Velocity is a vector quantity. Any change in its magnitude or direction or both causes acceleration.


Here, acceleration describes the rate of change of direction of the velocity.
3-31

You swing a 2.2 kg stone in a circle of radius 75 cm. At what speed should you swing it so its centripetal acceleration will be 9.8 m/s2?


Radius R = 75 cm = 0.75 m, centripetal acceleration .

and hence .
3-35

A wall clock has a second hand 15.0 cm long. What is the radial acceleration of the tip of this hand?


Radius R=0.150 m, T = 1 min = 60.0 s.


3-37

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very high 3.0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 cm from the axis of spin. What is the radial acceleration (in m/s2 and in g’s) of the endolymphfluid?
R = 0.070 m.

For 3.0 rev/s, the period T is



The force on the fluid must be 2.5 times its weight.









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