Math / Geometry

Question5) A rung on a hamster wheel, with a radius of 25 cm , is travelling at a constant speed. It makes one complete revolution in 3 seconds. The axle of the hamster wheel is 27 cm above the ground. a) Sketch a graph of the height of the rung above the ground during two complete revolutions, beginning when the rung is closest to the ground.

Studdy Solution

STEP 1

1. The hamster wheel is circular with a radius of $25$ cm.
2. The axle of the wheel is $27$ cm above the ground.
3. The wheel makes one complete revolution in $3$ seconds.
4. We are interested in the height of a specific rung above the ground.

STEP 2

1. Determine the initial position of the rung.
2. Calculate the height of the rung above the ground as a function of time.
3. Sketch the graph for two complete revolutions.

STEP 3

Determine the initial position of the rung:
- When the rung is closest to the ground, it is at the bottom of the wheel. - The distance from the axle to the ground is $27$ cm. - The distance from the axle to the bottom of the wheel is the radius, $25$ cm. - Therefore, the initial height of the rung above the ground is $27 - 25 = 2$ cm.

STEP 4

Calculate the height of the rung above the ground as a function of time:
- The height of the rung can be modeled as a sinusoidal function because it moves in a circular path. - The general form of the sinusoidal function is: $h(t) = A \sin(Bt + C) + D$ - Where: - $A$ is the amplitude, equal to the radius of the wheel, $25$ cm. - $B$ is related to the period of the function. Since the period is $3$ seconds, $B = \frac{2\pi}{3}$ . - $C$ is the phase shift. Since the rung starts at the minimum height, $C = -\frac{\pi}{2}$ . - $D$ is the vertical shift, equal to the axle height, $27$ cm.
- Therefore, the function becomes: $h(t) = 25 \sin\left(\frac{2\pi}{3}t - \frac{\pi}{2}\right) + 27$

STEP 5

Sketch the graph for two complete revolutions:
- The graph of $h(t)$ will be a sinusoidal wave. - The period of the function is $3$ seconds, so two complete revolutions will span $6$ seconds. - The minimum height is $2$ cm, and the maximum height is $52$ cm. - The graph starts at $2$ cm at $t = 0$ , rises to $52$ cm at $t = 1.5$ seconds, returns to $2$ cm at $t = 3$ seconds, rises again to $52$ cm at $t = 4.5$ seconds, and returns to $2$ cm at $t = 6$ seconds.
The graph is a sinusoidal wave with a period of $3$ seconds, starting at $2$ cm, peaking at $52$ cm, and repeating this pattern over $6$ seconds.

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