under certain conditions, the thrust T of a propeller varies jointly as the fourth power of it's diameter, d, and the square of the number, n, of revolutions per second. What happens to the thrust if n is doubled and d is halved?
T = kd^4n^2
if we replace n by 2n and d by d/2, we have
T' = k(d/2)^4(2n)^2 = kd^4n^2/4 = T/4
To understand what happens to the thrust when the variables are changed, we need to analyze the joint variation relationship given for the propeller.
According to the problem statement, the thrust T varies jointly as the fourth power of the diameter, d, and the square of the number of revolutions per second, n. Mathematically, this can be expressed as:
T ∝ d^4 * n^2
"∝" denotes proportionality.
Now, let's determine what happens to the thrust when n is doubled and d is halved. We can express these changes by multiplying n by 2 and dividing d by 2:
New thrust, T' ∝ (d/2)^4 * (2n)^2
Simplifying further, we have:
T' ∝ (1/16) * 4d^4 * 4n^2
T' ∝ (1/16) * 16 * d^4 * n^2
T' ∝ d^4 * n^2
Comparing this with the original equation for thrust, we see that the new thrust T' will be the same as the original thrust T. Therefore, if the number of revolutions per second is doubled and the diameter is halved, the thrust will remain the same.
In conclusion, the thrust will not change when doubling n and halving d under the given conditions.