Circular MotionDate: 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/ |
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