The Excel task Instructions

The data below were collected by performing a simple activity: measuring the centripetal force acting on a rotating object at different angular speeds.

The equation below shows the relationship between the force
(
𝐹
)
, mass
(
𝑚
)
, velocity
(
𝑣
)
and radius
(
𝑟
)
:

𝐹
=
𝑚
𝑣
2
𝑟

Where:

F = centripetal force (N)
m = mass (kg)
v = velocity (m/s)
r = radius of the circular path (m)

Transfer the following data to an Excel spreadsheet.

Velocity
(
𝑚
𝑠
−
1
)
Force
(
𝑁
)

0 0
0.5 0.14
1 0.61
1.5 1.42
2 2.47
2.5 3.93
3 5.66
3.5 7.68
4 9.87

In your Excel file, plot force
(
𝐹
)
on the y-axis and velocity
(
𝑣
)
on the x-axis.
Include the titles, axis labels, and correct units (3 marks).

Is the force proportional to velocity? Explain your observation in the Excel spreadsheet (1 mark).
Answer in your Excel spreadsheet using a text box.

Use the equation above to linearise the variables
𝐹
and
𝑣
to find the proportional relationship between
𝐹
and
𝑣
.
Add the proportional variables to the Excel spreadsheet table (1 mark).

Plot the linearised variables on a second graph. On this graph, make sure to put labels, units and a title.
Also include a line of best fit and the linear equation, as well as
𝑅
2
(2 marks).

What is the gradient value? (1 mark)
(Hint: use the second graph's equation to identify the value).
Answer in your Excel spreadsheet using a text box.

From the gradient value of your linear equation and the equation above, determine the radius of the circular path (r), making sure to write the units in your final answer (1 mark).
Assume the mass of the object is 0.5 kg
Answer in your Excel spreadsheet using a text box.

Note: Please use proper chart formatting, labels, a neat layout, and text boxes for comments and the final answer (1 mark).

Question

The Excel task Instructions

The data below were collected by performing a simple activity: measuring the centripetal force acting on a rotating object at different angular speeds.

The equation below shows the relationship between the force 
(
𝐹
)
, mass 
(
𝑚
)
 , velocity 
(
𝑣
)
 and radius 
(
𝑟
)
:

𝐹
=
𝑚
𝑣
2
𝑟

Where:

F = centripetal force (N)
m = mass (kg)
v = velocity (m/s)
r = radius of the circular path (m)

Transfer the following data to an Excel spreadsheet.

Velocity 
(
𝑚
𝑠
−
1
)
	Force 
(
𝑁
)

0	0
0.5	0.14
1	0.61
1.5	1.42
2	2.47
2.5	3.93
3	5.66
3.5	7.68
4	9.87

In your Excel file, plot force 
(
𝐹
)
 on the y-axis and velocity 
(
𝑣
)
 on the x-axis.
Include the titles, axis labels, and correct units (3 marks).

Is the force proportional to velocity? Explain your observation in the Excel spreadsheet (1 mark).
Answer in your Excel spreadsheet using a text box.

Use the equation above to linearise the variables 
𝐹
 and 
𝑣
 to find the proportional relationship between 
𝐹
 and 
𝑣
.
Add the proportional variables to the Excel spreadsheet table (1 mark).

Plot the linearised variables on a second graph. On this graph, make sure to put labels, units and a title.
Also include a line of best fit and the linear equation, as well as 
𝑅
2
  (2 marks).

What is the gradient value? (1 mark)
(Hint: use the second graph's equation to identify the value).
Answer in your Excel spreadsheet using a text box.

From the gradient value of your linear equation and the equation above, determine the radius of the circular path (r), making sure to write the units in your final answer (1 mark).
Assume the mass of the object is 0.5 kg 
Answer in your Excel spreadsheet using a text box.

Note: Please use proper chart formatting, labels, a neat layout, and text boxes for comments and the final answer (1 mark).

BlackTom AI · Accepted Answer

The task provides the centripetal force relationship F = m v^2 / r and asks to analyze data in Excel, including linearising F and v to find the proportional relationship.

First, recall the key equation: F = (m/r) * v^2. This means if you plot F on the y-axis against v^2 on the x-axis, the slope should be m/r. Equivalently, plotting F against v^2 will yield a straight line through the origin with gradient equal to m/r.

Next, consider the statement about linearising the variables F and v to find the proportional relationship between F and v. Linearising here means transforming the variables so that a linear fit can be applied. Since F is proportional to v^2, the natural linearisation is to plot F versus v^2 (or to plot F/(m) against v^2 to isolate r), and then extract the gradient which equals m/r.

Regarding the provided answer: 'After linearizing (Force vs Velocity²), the gradient from the linear equation is approximately 0.622 N/(m²/s²). Since F = (m/r)v², gradient = m/r. With mass m = 0.5 kg, the radius r = m/gradient ≈ 0.804 m (3 significant figures).'

Now, evaluate the logic step by step:
- The claim of linearising as Force vs Velocity² is appropriate for extracting the proportional relationship, because F ∝ v² when m and r are constants.
- The gradient of the F vs v² plot should indeed equal m/r, since F = (m/r) * v^2.
- Substituting m = 0.5 kg into r = m / gradient is correct algebra: r = m / (m/r) = r, so solving for r with the given gradient is the right approach.

Check for potential pitfalls or common misconceptions:
- If one incorrectly plots F against v, rather than F against v², the relationship would not be linear and the gradient would not correspond to m/r. This would lead to an erroneous r.
- The units must be consistent: F in newtons (N), m in kilograms (kg), v in meters per second (m/s), and r in meters (m). The gradient units N per (m²/s²) simplify to kg/m since N = kg·m/s² and v² = m²/s², giving (kg·m/s²) / (m²/s²) = kg/m, which matches the unit analysis for m/r when m is in kg and r in m.
- The calculation r = m / gradient uses the given gradient value; any rounding in the gradient will affect the final r. Using 0.622 with m = 0.5 kg yields r ≈ 0.804 m, which is consistent to three significant figures.

If you were to compute from the data directly:
- Determine F values for each v, then compute v² and fit F = (m/r) v² to obtain the slope. With m fixed at 0.5 kg, the slope gives r as r = m / slope.
- Ensure the Excel chart for the linearised graph includes the equation of the line and R² to verify the linear relationship; a good R² close to 1 indicates a strong linear fit.

In summary, the reasoning in the provided answer follows the correct method: linearise to F vs v², identify the gradient as m/r, and solve for r using the known mass. The numerical result of r ≈ 0.804 m is consistent with the given gradient 0.622 N/(m²/s²) and the stated mass.

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