Equation of the sinusoidal function that represents height above the ground

[NeomiXD]

Question:

1. A clock is hanging on wall. length of minute hand is 16cm and the length is 8cm for the hour hand. The highest that the tip of the minute hand reaches above ground is 265cm.

a) What is equation of axis, amplitude & period in minutes of function that represents the tip of hour hand's height above the ground.

b) Determine the equation of the sinusoidal function that represents the tip of the hour hand's height above the ground. Assume that at t = 0 min, the time is midnight.



So far I have done this:

a) max = 265cm

min = 265 - 2(16) = 233cm

amplitude = 16

c = 265 + 16 = 281cm

d = 0?

period = 60

b) y = 16cos(6x) +249

Are my answers right?

 

[cepheid]

Question:

1. A clock is hanging on wall. length of minute hand is 16cm and the length is 8cm for the hour hand. The highest that the tip of the minute hand reaches above ground is 265cm.

a) What is equation of axis, amplitude & period in minutes of function that represents the tip of minute hand's height above the ground.

b) Determine the equation of the sinusoidal function that represents the tip of the minute hand's height above the ground. Assume that at t = 0 min, the time is midnight.
So far I have done this:

a) max = 265cm

min = 265 - 2(16) = 233cm

amplitude = 16

Sure, looks alright.

c = 265 + 16 = 281cm

d = 0?

What are c and d supposed to represent physically? (I can't tell what you are trying to compute here).

period = ?? (how do you figure this out?)

The period is just the amount of time that it takes to do a full rotation. It's a minute hand. How long do you think it takes a minute hand to do a full rotation? The answer is really really obvious.

For this problem, it's much easier if you just draw a picture and do the actual trigonometry:

           /|
          / |
         /  | 
 r      /   |  h
       /    |  
      /     |
     /      |
    / θ     |
   ----------

The above is a picture of the minute hand at some arbitrary angle θ from the horizontal. The length of the hand is r (=16 cm). In this case, it's clear that the height of the tip of the hand (above the midpoint of the clock) is just h = rsinθ. (We've established that the midpoint of the clock is (233 + 16) cm above the ground, which means you just add this as a constant offset to your h value). So, how does h vary with time? To figure this out, we just need to know how θ varies with time. That's easy, because it's circular motion, and so the angular position of the minute hand is given by:

θ = θ₀ + ωt

where θ₀ is the initial angular position of the minute hand. In this case, it starts out vertical, so θ₀ = π/2. The angular velocity ω can be determined because you know the period (because you know it's a minute hand). Hence:

h = rsin(ωt + π/2)

and the height above the ground is h + 233 cm + 16 cm.

EDIT: note that sin(ωt + π/2) = cos(ωt)