Question

centripetal force (n) vs. speed m/s (grap...
centripetal force (n)
speed m/s
determine the relationship between the two variables on the graph from the choices above. match this with the actual variables in the centripetal force equation.

Explanation:

Step1: Recall Centripetal Force Formula

The centripetal force formula is \( F_c = \frac{mv^2}{r} \). If mass (\( m \)) and radius (\( r \)) are constant, \( F_c \) is proportional to \( v^2 \), but here the graph shows \( F_c \) (x - axis) and speed (\( v \), y - axis) with a negative slope? Wait, no, maybe misinterpretation. Wait, the x - axis is Centripetal Force (N), y - axis is Speed (m/s). Wait, the graph has a negative slope, so as \( F_c \) increases, speed decreases? But from \( F_c=\frac{mv^2}{r} \), if \( F_c \) increases, \( v \) should increase (for constant \( m,r \)). So maybe the other variable: if we consider \( F_c=\frac{mv^2}{r} \), rearranged as \( v = \sqrt{\frac{F_c r}{m}} \), but if \( r \) or \( m \) is changing. Wait, maybe the inverse: if we have \( F_c=\frac{m}{r}\times v^2 \), so \( F_c \) is proportional to \( v^2 \). But the graph here: let's check the axes. Wait, x - axis: Centripetal Force (0,100,200,300,400 N), y - axis: Speed (20,22,24,26,28 m/s). Wait, the points: when \( F_c = 200 \) N, speed is 20? No, wait the y - axis labels: 20,22,24,26,28 (from bottom to top? Wait, no, the y - axis is vertical, so lower y - values are at the bottom. Wait, the first blue dot: x = 200, y = 20? Then as x (F_c) increases, y (speed) increases? Wait, no, the line goes from (200,20) to (higher F_c, higher speed)? Wait, no, the x - axis is Centripetal Force (N), so 0,100,200,300,400 (left to right). The y - axis: 20,22,24,26,28 (bottom to top). So the first point is at x = 200, y = 20 (lower speed, lower F_c? No, x=200 is F_c=200 N, y=20 m/s. Then next point: x increases (F_c increases) and y increases (speed increases)? Wait, no, the line: let's see the coordinates. Wait, maybe I misread the axes. Wait, the title is "Centripetal Force (N) vs. Speed (m/s)". Wait, the x - axis is Centripetal Force, y - axis is Speed. Wait, the graph has a negative slope? Wait, no, the points: when F_c is 200, speed is 20; when F_c is higher (300,400), speed is higher (22,24,...)? Wait, no, the blue dots: let's list the points. Wait, the first dot: x≈200, y≈20; then x increases (towards 300,400) and y increases (towards 22,24,26,28). Wait, that would be a positive slope. Wait, maybe the x - axis is labeled wrong? Or maybe the formula is \( F_c=\frac{m}{r}\times v^2 \), so \( F_c \propto v^2 \), so the graph of \( F_c \) vs \( v \) would be a parabola, but here it's a straight line? Wait, no, the graph is a straight line, so linear relationship. Wait, maybe the inverse: \( v \propto \frac{1}{\sqrt{F_c}} \)? Then \( v = k/\sqrt{F_c} \), so as \( F_c \) increases, \( v \) decreases. But the graph here: as \( F_c \) increases (x - axis), \( v \) (y - axis) increases? Wait, no, the y - axis: 20 is at the bottom, 28 at the top. So when x (F_c) increases from 200 to 400, y (speed) increases from 20 to 28? That would be positive slope. But from \( F_c=\frac{mv^2}{r} \), \( F_c \) is proportional to \( v^2 \), so \( F_c = k v^2 \), so \( v=\sqrt{F_c/k} \), which is a square - root relationship, not linear. So maybe the other variable: if we consider \( F_c=\frac{mv^2}{r} \), then \( r=\frac{mv^2}{F_c} \), so \( r \) is proportional to \( v^2/F_c \). Or \( m=\frac{F_c r}{v^2} \). Wait, maybe the problem is to match the graph with the variables. The centripetal force equation is \( F_c=\frac{mv^2}{r} \), so the variables are \( F_c \) (centripetal force), \( m \) (mass), \( v \) (speed), \( r \) (radius). So the graph is between \( F_c \) and \( v \), so the relationship: from \( F_c=\frac{mv^2}{r} \), we can say \( v = \sqrt{\frac{F_c r}{m}} \), so \( v \) is proportional to \( \sqrt{F_c} \) (for constant \( m,r \)). But the graph here is linear, so maybe the square of speed vs \( F_c \) would be linear. Wait, maybe the y - axis is \( v^2 \) instead of \( v \), but the y - axis is labeled speed. Alternatively, maybe the inverse: if we have \( F_c=\frac{m}{r}\times v^2 \), so \( F_c \) vs \( v^2 \) is linear. But the graph here: x - axis \( F_c \), y - axis \( v \). So maybe the problem is to identify that the relationship between centripetal force and speed (from the equation \( F_c=\frac{mv^2}{r} \)) implies that centripetal force is proportional to the square of speed (for constant mass and radius). So if we plot \( F_c \) vs \( v^2 \), it's a straight line. But in this graph, y - axis is \( v \), x - axis \( F_c \), so the graph of \( F_c \) vs \( v \) would be a square - root curve, but the given graph is a straight line (maybe approximate or with other variables). Wait, maybe the key is to match the variables: the x - axis is centripetal force, y - axis is speed, and the equation is \( F_c=\frac{mv^2}{r} \), so the two variables are centripetal force (\( F_c \)) and speed (\( v \)), with mass (\( m \)) and radius (\( r \)) as constants (or one of them varying). So the relationship is that centripetal force is proportional to the square of speed (from \( F_c=\frac{mv^2}{r} \), so \( F_c \propto v^2 \) when \( m \) and \( r \) are constant).

Step2: Relate Graph to the Equation

The centripetal force equation is \( F_{c}=\frac{mv^{2}}{r} \). This shows that centripetal force (\( F_c \)) is directly proportional to the square of the speed (\( v \)) when the mass (\( m \)) and the radius (\( r \)) of the circular path are constant. In the given graph, the x - axis represents centripetal force (in Newtons) and the y - axis represents speed (in m/s). Even though the graph appears to have a linear trend (from the plotted points), the underlying relationship from the centripetal force formula is that \( F_c \) is proportional to \( v^2 \) (for constant \( m \) and \( r \)). So the two variables in the graph (centripetal force and speed) are related by the centripetal force formula, where centripetal force depends on the square of the speed (along with mass and radius).

Answer:

The relationship between the two variables (centripetal force and speed) is described by the centripetal force formula \( \boldsymbol{F_{c}=\frac{mv^{2}}{r}} \), where centripetal force (\( F_c \)) is proportional to the square of the speed (\( v \)) (for constant mass \( m \) and radius \( r \) of the circular path). The graph's variables (centripetal force on the x - axis, speed on the y - axis) are related through this equation, with centripetal force depending on the square of the speed (along with mass and radius).