an "extended object" is in balance when the center of mass is located directly over the fulcrum. By using a test mass mT placed at various points xT on the object (as measured from some fixed reference point), a linear relationship between xT and the fulcrum location xF is found (equation 6):
m_(T)x_(T) = (M+m_(T))x_(F) - MX_(CM)
If a plot of mTxT versus xF were made from an object of M = 125 g with XCM = 4 cm, using a test mass of mT = 30 g, what would be the slope and y-intercept of the line? Your answers will need units.
Slope:
Y-intercept:
To find the slope and y-intercept of the line, we need to rearrange the equation and compare it with the standard form of a linear equation, y = mx + b.
Given the equation:
m_(T)x_(T) = (M + m_(T))x_(F) - MX_(CM)
We can rewrite it as:
mTxT = (M + mT)xF - MXCM
Comparing this with y = mx + b, we can identify the slope (m) and y-intercept (b):
Slope (m) is given by:
m = (M + mT)
Y-intercept (b) is given by:
b = -MXCM
In this case, M = 125 g, XCM = 4 cm, and mT = 30 g. Let's substitute these values into the equations to calculate the slope and y-intercept:
Slope:
m = (M + mT)
= (125 g + 30 g)
= 155 g
Y-intercept:
b = -MXCM
= -(125 g * 4 cm)
= -500 g·cm
Therefore, the slope of the line is 155 g and the y-intercept is -500 g·cm.