On your first day at work for an appliance manufacturer, you are told to figure out what to do to the period of rotation during a washer spin cycle to triple the centripetal acceleration. You impress your boss by answering immediately.
Express in terms of T.
Equation I believe that needs to be used is a = 4π²r/T²?
To triple the centripetal acceleration during a washer spin cycle, we can use the equation you provided: a = (4π²r)/T².
Let's break down the equation:
a: Centripetal acceleration (the acceleration towards the center of the circular motion)
r: Radius of rotation
T: Period of rotation (the time taken for one complete revolution)
To triple the centripetal acceleration, we need to find the new period of rotation, T'.
First, let's express the initial centripetal acceleration, a1, in terms of T:
a1 = (4π²r)/T²
Now, let's find the new centripetal acceleration, a2 (which is three times the initial centripetal acceleration):
a2 = 3a1
= 3[(4π²r)/T²]
To find the new period of rotation, T', we rearrange the equation:
a2 = (4π²r)/T'²
Substituting the expression for a2:
3[(4π²r)/T²] = (4π²r)/T'²
To simplify further, we can cancel out the common terms:
3/T² = 1/T'²
Now, solve for T':
T'² = T²/3
Finally, take the square root of both sides:
T' = √(T²/3)
Therefore, to triple the centripetal acceleration during the washer spin cycle, you need to reduce the period of rotation to √(T²/3).
To determine what needs to be done to the period of rotation (T) in order to triple the centripetal acceleration (a), we can start by using the correct equation for centripetal acceleration:
a = (4π²r) / T²
Where:
a is the centripetal acceleration
r is the radius of rotation
T is the period of rotation
Now, our goal is to triple the centripetal acceleration, so we need to find out what needs to happen to T in order to achieve this. Let's go step by step:
1. Triple the centripetal acceleration:
To triple a, we need to multiply it by 3. So the new centripetal acceleration will be 3a.
2. Substitute the new value of a into the equation:
Replacing a with 3a in the equation, we get:
3a = (4π²r) / T²
3. Solve for T:
To find what needs to be done to T, we want to isolate it on one side of the equation. Let's rearrange the equation:
3a = (4π²r) / T²
Multiplying both sides of the equation by T², we get:
3a(T²) = 4π²r
Now, divide both sides of the equation by 3a:
T² = (4π²r) / (3a)
Finally, take the square root of both sides to solve for T:
T = √((4π²r) / (3a))
And that's it! The formula T = √((4π²r) / (3a)) gives us the period of rotation (T) that would triple the centripetal acceleration (a), expressed in terms of r and a.
a = v^2/r
v T = 2 pi r
so
v = 2 pi r/T
v^2 = (2 pi)^2 r^2/T^2
a = v^2/r = (2 pi)^2 r /T^2
divide T by sqrt 3
a' = (2 pi)^2 r / (T^2/3)