the speed of a gear wheel varies inversely as the number of teeth of the gear. if a gear wheel which has 48 teeth makes 3 revolutions per minute. how many revolutions per minute will a gear wheel with 96 teeth make?
twice the teeth means half the revolutions
teeth * speed = k = 48 * 3 = 144
96 * s = 144 ... s = 144 / 98
To solve this problem, we can set up a proportion using the inverse variation relationship between the speed of the gear wheel and the number of teeth.
Let's denote the speed of the gear wheel as S and the number of teeth as T.
According to the information given, we have:
S ∝ 1/T
Also, we know that when T = 48, S = 3.
48 × S = 1 × 3
48S = 3
S = 3/48
S = 1/16
Now, we can use the constant of variation to find the speed of a gear wheel with 96 teeth.
96 × S = 1 × R
96 × (1/16) = R
6 = R
Therefore, a gear wheel with 96 teeth will make 6 revolutions per minute.
To find the number of revolutions per minute that a gear wheel with 96 teeth will make, we can use the concept of inverse variation.
Let's first understand what inverse proportion means in this context. When two quantities are said to vary inversely, it means that as one quantity increases, the other decreases, and vice versa. In this case, the speed of the gear wheel is inversely proportional to the number of teeth.
We are given that a gear wheel with 48 teeth makes 3 revolutions per minute. So, if the number of teeth in a gear wheel doubles (from 48 to 96), we can expect the number of revolutions per minute to decrease by half.
To calculate the new number of revolutions per minute, we can set up an inverse variation equation:
(speed of gear with 48 teeth) * (number of teeth of gear with 48 teeth) = (speed of gear with 96 teeth) * (number of teeth of gear with 96 teeth)
Let's represent the speed of the gear with 48 teeth as "s1" and the speed of the gear with 96 teeth as "s2". Similarly, let's represent the number of teeth of the gear with 48 teeth as "t1" and the number of teeth of the gear with 96 teeth as "t2".
s1 * t1 = s2 * t2
From the given information, we know that s1 = 3 and t1 = 48. We need to solve for s2.
3 * 48 = s2 * 96
144 = 96s2
Dividing both sides by 96:
s2 = 144 / 96
s2 = 1.5
Therefore, a gear wheel with 96 teeth will make 1.5 revolutions per minute.