6) The relationship between force and acceleration is given by Newton’s 2nd Law: Fnet = ma. In this case this becomes:
Fapp – Ff = ma.
a. Rearrange this formula to fit the equation of a straight line:
y = mx + b
b. What does the y‐intercept represent?
c. What does the slope represent?
a. To rearrange the formula Fapp - Ff = ma to fit the equation of a straight line (y = mx + b), we can assign variables as follows:
- Let Fapp - Ff = y
- Let m = a
- Let x = Fapp
Therefore, the rearranged formula is y = mx + b, where y represents Fapp - Ff, m represents acceleration, x represents Fapp, and b represents the y-intercept.
b. The y-intercept (b) represents the value of y (Fapp - Ff) when x (Fapp) is zero. In the context of the formula Fapp - Ff = ma, when there is no applied force (Fapp = 0), the only force acting on the object is the force of friction (Ff). Therefore, the y-intercept represents the force of friction (Ff) in this scenario.
c. The slope (m) represents the relationship between the variables x (Fapp) and y (Fapp - Ff). In the context of the formula Fapp - Ff = ma, the slope represents the mass (m) of the object. It indicates the amount of force (Fapp) needed to accelerate the object (ma). A larger mass (m) would result in a steeper slope, indicating that more force is required to produce the same acceleration.
a. To rearrange the formula Fapp - Ff = ma to fit the equation of a straight line y = mx + b, we need to isolate an acceleration term. We can start by moving the Ff term to the other side of the equation:
Fapp = Ff + ma
Next, we can rewrite the equation in terms of a:
Fapp - Ff = ma
Now, to fit the equation y = mx + b, we can consider the following equivalences:
y represents Fapp - Ff
m represents the slope, which in this case, is equal to a
x can represent the mass of the object
b represents the y-intercept, which is the value of y when x (or mass) is zero.
Thus, we rearrange the formula as:
y = ax + b
b. In the context of the rearranged equation, the y-intercept (b) represents the value of y (Fapp - Ff) when x (mass) is equal to zero. This means that the y-intercept represents the force applied (Fapp) on an object when there is no friction (Ff) and no mass (or acceleration) involved. It indicates the initial force acting on the object.
c. The slope (m) in this context represents the acceleration (a). The slope of a straight line equation represents the rate of change of y with respect to x. In this case, it indicates how the net force on an object (Fapp - Ff) changes with respect to the mass (x). A steeper positive slope implies a higher acceleration, while a negative slope would represent deceleration or slowing down of the object.
b) Ff
c) m