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!
Well I looked in the back of my book, and for a, the answer is shown as 3.1 mm/s.
Not sure what I did wrong. Sorry!
Well, my assumption that each hand is rotating one hour may be incorrect. How far is each hand rotating?
Does not give me that information... what i typed was what it gave me. :(
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
To find the linear velocity of each hand of a clock, we can use the formula:
Linear Velocity = Circumference of the Circle / Time taken for one complete revolution
a. For the second hand:
The second hand completes one full revolution in 60 seconds and its length is given as 30 millimeters. To find the circumference of the circle, we use the formula for circumference of a circle:
Circumference = 2 * π * radius
Since the radius is equal to the length of the second hand, the circumference is:
Circumference = 2 * π * 30 = 60π mm
Therefore, the linear velocity of the second hand is:
Linear Velocity = 60π mm / 60 seconds = π mm/s
b. For the minute hand:
The minute hand completes one full revolution in 60 minutes (or 3600 seconds) and its length is given as 27 millimeters. Using the same formula, the circumference is:
Circumference = 2 * π * 27 = 54π mm
Therefore, the linear velocity of the minute hand is:
Linear Velocity = 54π mm / 3600 seconds = π / 66 mm/s
c. For the hour hand:
The hour hand completes one full revolution in 12 hours (or 43,200 seconds) and its length is given as 18 millimeters. Using the formula, the circumference is:
Circumference = 2 * π * 18 = 36π mm
Therefore, the linear velocity of the hour hand is:
Linear Velocity = 36π mm / 43,200 seconds = π / 1,200 mm/s
In summary:
a. The linear velocity of the second hand is π mm/s.
b. The linear velocity of the minute hand is π / 66 mm/s.
c. The linear velocity of the hour hand is π / 1,200 mm/s.
Please note that these values are approximations since we are assuming a perfectly circular motion of the clock hands.
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.