Tuning Forks

Tuning Instruments

Tag: tuning

tuning fork c

tuning fork c

Trig questions involving sine and cosine?

please explain how to them and pls. the complete answers as well.

1) A reflector is fastened to the front wheel of a bicycle 20 cm from the center of the wheel. The diameter of the wheel and inflated tire is 70 cm.
The bike is traveling at 10km/hr.

a) Express the height of the reflector above the ground as a function of time. Assume that at t=0 seconds, the reflector is at its highest point.

b) At what time will the reflector be at its lowest point.

3) A tuning fork is a small steel instrument used to tune pianos and test hearing. A C tuning fork vibrates at 264 hz, with an amplitude of .02 cm. Write the equation of the motion of the fork.

Okay, for part one we first need to find out how long it takes for the wheel to rotate.

The wheel has a diameter of 70cm. The circumference of the wheel is found through the equation C = d*pi. Therefore, the circumference is 70pi cm.

Now that we have the length of the part of the wheel that touches the ground, let’s find out how long it takes it to make a full rotation. Assuming every cm rotated is a cm traveled, divide the circumference by the velocity of the bike.
70pi / (10km/hr)
70pi / (10km/hr * 100000cm/km) conversion of km to cm
70pi / (1000000cm/hr)
pi*7.0×10^-5 hrs for a rotation.

Since we are dealing with seconds, convert this figure to seconds.

pi*4.2×10^-3 sec. for a rotation

Now, since the reflector will take this same amount of time to fully revolve about the center, it will use the same time.

A full revolution, in radians, is calculated as 2pi. Since it takes pi*4.2×10^-3 sec for a revolution, the rate is:

2pi/(pi*4.2×10^-3) rev/sec, or, simplified:
1/(2.1×10^-3) rev/sec, which further simplifies to:
2.1×10^3 rev/sec, due to the neg. exponent.

Now we can create the formula for the reflector.

We are going to use sine, as sine measures the y axis. To visualize this, image the radian circle. At pi/2, the graph is straight upward and sine is measured at its highest value at that point.

So, r = sin(t*speed) is the formula we have so far, where t is equal to the time elapsed in seconds and speed is the rev/sec figure we calculated. But, alas, just the portion alone is not complete. If we kept it at this, when the quantity that sine is evaluating is in the lower region, it will be negative. This is because we have not factored in height. Let’s find the lowest height of reflector.

At the lowest height, the reflector is straight down. This means that the reflector is at a length of the radius, minus its distance from the center. To get the radius, divide the diameter in half to get 35cm. This means, that at its lowest point, the reflector is 35 – 20 cm from the bottom, or 15cm from the bottom. Do not factor this in yet.

Now let us determine the maximum height. This will be the radius plus the distance of the reflector from the center. 35 + 20 = 55cm.

So, the reflector will range from 15cm to 55cm, a difference of 40.

Now let’s conceptualize here. If sine produces +1 at the maximum height, and -1 at the lowest, then we need to find the middle point between the two heights to counteract this negation. 15 + 55 = 70. 70/2 = 35. The center will be measured as 35. So, recap. The reflector will differ its position by 20, as that is the distance from the center. And the height of the center is 35. Therefore, the formula will look something like this.
r = 20sin((2.1×10^3)*t) + 35

Now, I am not going to solve the rest, but I hope this explanation will serve as a visual catalyst, and help you to complete the problem.