A rectangularplate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner, as the drawing shows.The centripetal acceleration measured atcorner A is n times as greatas that measured at corner B. What is theratio
L1
L2
of thelengths of the sides of the rectanglewhen n = 2.04?
L1
L2
= ?
To solve this problem, we need to analyze the physics of the rotating plate and understand the concept of centripetal acceleration.
First, let's assume that the length of one side of the rectangle is L1 and the length of the adjacent side is L2. Since the plate is rotating with a constant angular speed, it has a constant angular velocity. We can assume that the corner A is at a distance R from the axis of rotation, where R is the distance from the axis to the corner of the rectangle.
Centripetal acceleration is the acceleration directed towards the center of the circular path. In this case, the centripetal acceleration at corner A is given by:
a_A = ω^2 * R,
where ω is the angular velocity.
Similarly, the centripetal acceleration at corner B, which is at a distance L2 from the axis, is given by:
a_B = ω^2 * L2.
We are given that the centripetal acceleration at corner A is n times as great as that at corner B:
a_A = n * a_B.
Substituting the expressions for a_A and a_B, we get:
ω^2 * R = n * ω^2 * L2.
Simplifying the equation, we find:
R = n * L2.
Since we know that the diagonal of the rectangle is equal to R, we can use the Pythagorean theorem to find the ratio L1/L2:
L1^2 + L2^2 = R^2.
Since we know that R = n * L2, we can substitute it into the equation:
L1^2 + L2^2 = (n * L2)^2.
Simplifying the equation, we get:
L1^2 = L2^2 * (n^2 - 1).
Taking the square root of both sides, we find:
L1 = L2 * sqrt(n^2 - 1).
Now, we can substitute n = 2.04 into the equation to find the ratio L1/L2:
L1/L2 = sqrt((2.04)^2 - 1).
Calculating sqrt((2.04)^2 - 1), we get approximately 2.037.
Therefore, the ratio L1/L2 is approximately 2.037 when n = 2.04.