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Circular Motion


Date: 01/27/2001 at 17:58:27
From: Kim
Subject: Trigonometry word problems involving speed and angular speed.

I have two questions. I can't find any formula in my book to help me, 
and I don't know where to begin

1) An electric hoist is being used to lift a piece of equipment. The 
radius of the drum on the hoist is 10 inches, and the equipment must 
be raised 1 foot. Find the number of degrees through which the drum 
must rotate.

2) A car is moving at a rate of 50 miles per hour and the radius of 
its wheels is 2.5 feet. A) find the number of revolutions per minute 
the wheels are rotating. B) Find the angular speed of the wheels in 
radians per minute.

I have the formula for speed and angular speed, I just don't know 
where the values in the problem would fall.

Thank you for any help you can offer, 
Kim


Date: 01/28/2001 at 13:16:56
From: Doctor Roy
Subject: Re: Trigonometry word problems involving speed and angular 
speed.

Hello,

Thanks for writing to Dr. Math.  

Let's recall some facts about circular motion for each of these 
problems.

1) We must raise an object 1 ft. The pulley drum has a radius of 10 
in. In other words, we must move the rope along the pulley drum 1 ft 
(= 12 in). 

We can find that the circumference of a circle is 2*pi*r, which is 20*
pi in. in this case. So, we rotate the pulley by 12/(20*pi). We know 
that a circle has 2*pi radians. We now have a proportion:

       12/(20*pi) = x/(2*pi), where x is the angle through which the 
                              pulley goes.

                x = 12*(2*pi) / (20*pi)
                  = ???? radians

We can now convert this to angular degrees.

2) We have a car traveling 50 miles an hour with a tire radius of 
2.5 ft.

a) We wish to find the rotational velocity of the tires in revolutions 
per minute.

First let's convert the car's speed to ft/min, since this will 
probably be more convenient.

        50 miles/hour * (5280 ft/mile) * (1 hr/60 min)

              =  4400 ft/min

We know that this is some multiple of the number of circumferences of 
the tire (since the tire rolls smoothly along). Let's find out how 
many circumferences (or revolutions) this is:

            C = 2*pi*r

              = 2*pi*2.5 ft

              = 5*pi ft

   So, there are 5*pi ft per revolution of the tires.

   Let's convert this speed to revolutions per minute:

          (4400 ft/min) / (5*pi ft/rev)

            = ???? revolutions per min.

b) We wish to convert this measurement to angular speed in radians per
minute. This is a simple conversion from revolutions to radians.

Recall that there are 2*pi radians per revolution. If x is our speed 
in revolutions per minute, we can find the speed in radians per minute 
by the following:

          (x rev/min) * (2*pi radians / rev)

            = 2*pi*x radians/min

And we have our solution.

I hope this helps. Feel free to write again.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Trigonometry

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