Question

scis mechanical systems | assgt5 - speed ratio
name:
class:
write the three versions of the speed ratio formula, isolating each of the three variables:

Explanation:

Step1: Recall Speed Ratio Formula

The speed ratio (SR) for mechanical systems (e.g., gear trains, pulley systems) is typically defined as the ratio of the number of teeth (or revolutions, or diameters) of the driven component to the driving component. Let's denote:

• \( N_1 \): Speed (revolutions per unit time) of the driving component
• \( N_2 \): Speed of the driven component
• \( T_1 \): Number of teeth (or diameter, etc.) of the driving component
• \( T_2 \): Number of teeth of the driven component

The basic speed ratio formula is \( SR = \frac{N_2}{N_1} = \frac{T_1}{T_2} \) (since speed is inversely proportional to the number of teeth for gear systems: more teeth → slower speed).

Step2: Isolate \( N_2 \)

Start with \( SR = \frac{N_2}{N_1} \). Multiply both sides by \( N_1 \):
\( N_2 = SR \times N_1 \)

Step3: Isolate \( N_1 \)

From \( SR = \frac{N_2}{N_1} \), rearrange to solve for \( N_1 \). Multiply both sides by \( N_1 \), then divide by \( SR \):
\( N_1 = \frac{N_2}{SR} \)

Step4: Isolate \( SR \)

The original formula already isolates \( SR \):
\( SR = \frac{N_2}{N_1} \) (or using teeth: \( SR = \frac{T_1}{T_2} \), but for speed variables, \( SR = \frac{N_2}{N_1} \))

Answer:

1. Isolating \( N_2 \): \( \boldsymbol{N_2 = SR \times N_1} \)
2. Isolating \( N_1 \): \( \boldsymbol{N_1 = \frac{N_2}{SR}} \)
3. Isolating \( SR \): \( \boldsymbol{SR = \frac{N_2}{N_1}} \)

(Note: If using teeth/diameter (\( T_1, T_2 \)) instead of speed (\( N_1, N_2 \)), the formulas would be \( T_2 = \frac{T_1}{SR} \), \( T_1 = SR \times T_2 \), \( SR = \frac{T_1}{T_2} \), but the problem specifies "speed ratio formula" with speed variables, so the above uses \( N_1, N_2 \).)