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PreCalc

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Consider the tip of each hand of a clock. Find the linear velocity in millimeters per second for each hand.

a. second hand which is 30 millimeters.

b. minute hand which is 27 millimeters long.

c. hour hand which is 18 millimeters long.

Please tell me how you got the answer. Thanks!

  • PreCalc -

    Linear velocity equals angular velocity (in radians) multiplied by radius.

    Let's start by calculating angular velocity. Angular velocity is a measure of the angular displacement per unit time. (The angular velocity of a particle traveling on a circular path is the ratio of the angle traversed to the amount of time it takes to traverse that angle.) w = angle/time

    From what you've told me, I'm going to assume we are rotating one hour, which is an entire circle. That is 360 degrees, or 2(pi) radians. Note that one hour is 3600 seconds, and the length of each hand is the radius.

    a. w = angle/time = 2(pi) / 3600 = (pi)/1800.
    Linear velocity is that times radius. (pi/1800)*30 = 30(pi)/1800 = pi/60 mm/sec

    Do a similar process for b and c, and I'll critique your thinking. If you have any questions, let me know.

  • PreCalc -

    Well I looked in the back of my book, and for a, the answer is shown as 3.1 mm/s.

  • PreCalc -

    Well, my assumption that each hand is rotating one hour may be incorrect. How far is each hand rotating?

  • PreCalc -

    Does not give me that information... what i typed was what it gave me. :(

  • PreCalc -

    Not sure what I did wrong. Sorry!

  • PreCalc -

    a) for the second hand..

    it rotates 2pi rad/min
    =2pi/60 rad/sec
    = pi/30 rad/sec and it is 30 mm long

    so the linear velocity = (pi/30)(30 mm/sec
    = pi mm/sec
    = 3.14159 mm/sec (you book had 3.1)

    b) for the minute hand
    it rotates 2pi radians per hour
    = 2pi/3600 rad/sec

    so linear vel. = pi/1800)(27) mm/sec
    = appr .048 mm/sec

    c) the hour hand rotates 2pi radians each 12 hours

    so angular vel. = 2pi/(43200) rad/sec

    linear vel. = 18(2pi/43200) mm/sec
    = appr. .00262 mm/sec

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