If gear A has 50 teeth and runs at 40 rpm and is used in driving gear B with 25 teeth, at what speed will gear B be moving?
Answer (2)
The speed of gear B is calculated to be 80 rpm using the relationship between the teeth of the gears and their speeds. By substituting the values into the formula, we find that gear B rotates faster due to having fewer teeth than gear A. Therefore, gear B responds to gear A's rotation by turning at a speed of 80 rpm.
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Establish the relationship between the number of teeth and speed of gears: T A S A = T B S B .
Substitute given values: 50 × 40 = 25 × S B .
Solve for the speed of gear B: S B = 25 50 × 40 = 80 .
The speed of gear B is 80 rpm.
Explanation
Problem Analysis Let's analyze the problem. We have two gears, A and B, connected in such a way that gear A drives gear B. We know the number of teeth and the speed of gear A, and the number of teeth of gear B. We need to find the speed of gear B.
Formula Introduction The relationship between the number of teeth and the speed of two gears is given by the formula: T A S A = T B S B where:
T A is the number of teeth of gear A,
S A is the speed of gear A in rpm,
T B is the number of teeth of gear B,
S B is the speed of gear B in rpm.
Given Values We are given:
T A = 50 teeth
S A = 40 rpm
T B = 25 teeth We need to find S B .
Substitution Substitute the given values into the formula: 50 × 40 = 25 × S B
Calculation Solve for S B :
S B = 25 50 × 40 S B = 25 2000 S B = 80
Final Answer Therefore, the speed of gear B is 80 rpm.
Examples
Understanding gear ratios is crucial in many mechanical systems, such as bicycles and cars. For example, in a bicycle, the gear ratio determines how many times the rear wheel rotates for each rotation of the pedals. If a bicycle has a gear ratio of 3:1, it means the rear wheel rotates three times for every rotation of the pedals. This concept helps cyclists adjust the difficulty of pedaling based on the terrain. Similarly, in a car, different gear ratios are used to optimize engine performance and fuel efficiency at various speeds.
