A rectangular plate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner, as the drawing shows. The centripetal acceleration measured at corner A is n times as great as that measured at corner B. What is the ratio L1/L2 of the lengths of the sides of the rectangle when n = 2.80?
I did sqrt 2.79 = 1.67 and then 1/1.67=0.59
Is this correct?
This is how the solution is reached:
Centripital acceleration = radius x (angular speed)^2
The acceleration of A = 2.8 (the acceleration of B)
So to set it up we get:
Sqrt(L2^2+L1^2) w^2 = 2.8 (L1) w^2
The angular velocities cancel because they are equal.
Now square both sides to get:
(L2^2+L1^2) = 7.84 (L1)^2
Move the L1 over:
(L2^2+L1^2)/L1^2=7.84
The fraction can be written:
L2^2/L1^2 + L1^2/L1^2 = 7.8
Simplify:
L2^2/L1^2 + 1= 7.84
L2^2/L1^2= 6.84
Take the sqrt of that:
L2/L1= 2.615
The reciprocal of that is 1/2.615
L1/L2= .382
Well, my calculations may not always be the most reliable, but I can still attempt to give you a laugh! Let's see what I can come up with.
Ahem, ahem. Why did the rectangle go on a diet? Because it wanted to achieve the perfect L1/L2 ratio, of course!
Now, coming back to your question, it seems like you're on the right track! However, your calculations might have been a bit off. To find the ratio L1/L2, we need to take the square root of n (which is given as 2.80) to get the ratio of centripetal accelerations. So, √2.80 ≈ 1.67.
But when it comes to finding the ratio of lengths, it's actually the reciprocal of this value. So, 1/1.67 ≈ 0.60. Therefore, the correct ratio of L1/L2 when n = 2.80 is approximately 0.60.
I hope my little clownish explanation was able to provide some amusement while still being informative. Keep on smiling and asking questions!
To find the ratio of the lengths of the sides of the rectangle (L1/L2) when the centripetal acceleration at corner A is n times greater than at corner B, you have correctly determined the value 1.67 as the square root of 2.80.
To find L1/L2, you need to divide the length of one side by the length of the other side.
Let's assume L1 represents the longer side and L2 represents the shorter side.
Using the given value of n = 2.80, the centripetal accelerations at corners A and B are related as follows:
(aA / aB) = n
Since the centripetal acceleration is given by a = ω^2 * r, where ω is the angular speed and r is the distance to the axis of rotation, we have:
(ω^2 * rA) / (ω^2 * rB) = n
Here, rA is the distance between corner A and the axis of rotation, and rB is the distance between corner B and the axis of rotation.
Since the axis of rotation passes perpendicularly through corner A, rA is the length of the longer side L1, and rB is the length of the shorter side L2.
Thus, we have:
(L1 / L2) = √(rA / rB) = √(n)
Substituting n = 2.80, we have:
(L1 / L2) = √(2.80) ≈ 1.674
Therefore, the correct ratio of the lengths of the sides when n = 2.80 is approximately 1.674, which differs slightly from your calculated value of 1.67.
To find the ratio of the lengths of the sides of the rectangle, L1/L2, we need to use the given information about the centripetal accelerations measured at corners A and B.
The formula for centripetal acceleration is:
ac = ω^2 * r,
where ac is the centripetal acceleration, ω is the angular speed, and r is the distance from the axis of rotation to the point at which the acceleration is measured.
In this case, we are given that the centripetal acceleration at corner A is n times greater than the centripetal acceleration at corner B. Therefore, we can write:
acA = n * acB.
Using the formula for centripetal acceleration, we can express acA and acB in terms of the angular speed ω and the distances from the axis of rotation to the respective corners:
ω^2 * rA = n * (ω^2 * rB).
Since the angular speed ω is constant, we can cancel it out from both sides of the equation:
rA = n * rB.
Now, we can use the fact that the rectangle is rotating about an axis that passes perpendicularly through one corner. Let's assume that side L1 of the rectangle is adjacent to corner A and L2 is adjacent to corner B.
Since the axis passes through corner A, the distance from the axis of rotation to corner A, rA, is equal to L1. Similarly, the distance from the axis of rotation to corner B, rB, is equal to L2.
Thus, we can rewrite the equation above as:
L1 = n * L2.
Now, we can solve for the ratio L1/L2 when n = 2.80:
L1/L2 = n.
Therefore, when n = 2.80, the ratio L1/L2 is simply 2.80.
So, your calculation of L1/L2 as 0.59 is incorrect. The correct ratio is 2.80.
Let the point where the axis passes through the plate be O.
OA =sqrt (L1²+L2²),
OB = L1
a(centripetal) = ω²•R,
a(A) =ω²•OA= ω²•sqrt (L1²+L2²),
a(B) = ω²•OB = ω²•L1.
a(A)/a(B) = ω²•sqrt(L1²+L2²)/ω²•L1 = n,
n= sqrt (L1²+L2²)/ L1,
n²=(L1²+L2²)/(L1)² =1+(L2/L1)²
(L2/L1)² = n²-1,
L2/L1 = sqrt(n²-1),
L1/L2=1/ sqrt(n²-1) = =1/sqrt(2.8²-1)=3.8.