Tool 3: Mathematics

Estimate the acceleration of the bottle when it is at its maximum height.

Estimate the acceleration of the bottle when it is at its maximum height.

Estimate the acceleration of the bottle when it is at its maximum height.

Estimate, using the graph, the maximum height of the bottle.

Estimate, using the graph, the maximum height of the bottle.

Estimate, using the graph, the maximum height of the bottle.

Determine, using the graph, whether the gas acts as an ideal gas.

Determine, using the graph, whether the gas acts as an ideal gas.

Determine, using the graph, whether the gas acts as an ideal gas.

Determine, using the graph, whether the gas acts as an ideal gas.

Determine, using the graph, whether the gas acts as an ideal gas.

Determine, using the graph, whether the gas acts as an ideal gas.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

State the unit for k.

State the unit for k.

State the unit for k.

State the unit for k.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

A rectangular sheet of paper has dimensions of (30.0 ± 0.5) cm and (20.0 ± 0.5) cm.

What is the percentage uncertainty of the perimeter of the paper?

A.  1 %

B.  2 %

C.  2.5 %

D.  4 %

A rectangular sheet of paper has dimensions of (30.0 ± 0.5) cm and (20.0 ± 0.5) cm.

What is the percentage uncertainty of the perimeter of the paper?

A.  1 %

B.  2 %

C.  2.5 %

D.  4 %

Determine the fundamental SI unit for a.

Determine the fundamental SI unit for a.

Determine the fundamental SI unit for a.

Determine the fundamental SI unit for a.

Determine the fundamental SI unit for a.

Determine the fundamental SI unit for a.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Suggest an appropriate measuring instrument for determining b.

Suggest an appropriate measuring instrument for determining b.

Suggest an appropriate measuring instrument for determining b.

Suggest an appropriate measuring instrument for determining b.

Suggest an appropriate measuring instrument for determining b.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Calculate the largest fractional uncertainty in T for these data.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Calculate the percentage uncertainty in the value of A.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Determine A by drawing the line of best fit.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of $E$ are kg m⁻¹s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of $E$ are kg m⁻¹s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of $E$ are kg m⁻¹s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of $E$ are kg m⁻¹s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of $E$ are kg m⁻¹s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

Theory predicts that

$d\propto \frac{W^x{L}^y}{EI}$

where $E$ and $I$ are constants. The fundamental units of $I$ are m⁴ and those of E are kg m⁻¹ s⁻².

Calculate $x$ and $y$.

A student is verifying the equation

$$x=\frac{2\lambda Y}{z}$$

The percentage uncertainties are:

What is the percentage uncertainty in x?

A. 5 %

B. 15 %

C. 25 %

D. 30 %

A student is verifying the equation

$$x=\frac{2\lambda Y}{z}$$

The percentage uncertainties are:

What is the percentage uncertainty in x?

A. 5 %

B. 15 %

C. 25 %

D. 30 %

In an experiment to determine the resistivity of a material, a student measures the resistance of several wires made from the pure material. The wires have the same length but different diameters.

Which quantities should the student plot on the $x$-axis and the $y$-axis of a graph to obtain a straight line?

In an experiment to determine the resistivity of a material, a student measures the resistance of several wires made from the pure material. The wires have the same length but different diameters.

Which quantities should the student plot on the $x$-axis and the $y$-axis of a graph to obtain a straight line?

A student models the relationship between the pressure p of a gas and its temperature T as p = $x$ + $y$T.

The units of p are pascal and the units of T are kelvin. What are the fundamental SI units of $x$ and $y$?

A student models the relationship between the pressure p of a gas and its temperature T as p = $x$ + $y$T.

The units of p are pascal and the units of T are kelvin. What are the fundamental SI units of $x$ and $y$?

A student measures the radius R of a circular plate to determine its area. The absolute uncertainty in R is ΔR.

What is the fractional uncertainty in the area of the plate?

A. $\frac{2\mathrm{\Delta }R}{R}$

B. ${\left(\frac{\mathrm{\Delta }R}{R}\right)}^2$

C. $\frac{2\pi \mathrm{\Delta }R}{R}$

D. $\pi {\left(\frac{\mathrm{\Delta }R}{R}\right)}^2$

A student measures the radius R of a circular plate to determine its area. The absolute uncertainty in R is ΔR.

What is the fractional uncertainty in the area of the plate?

A. $\frac{2\mathrm{\Delta }R}{R}$

B. ${\left(\frac{\mathrm{\Delta }R}{R}\right)}^2$

C. $\frac{2\pi \mathrm{\Delta }R}{R}$

D. $\pi {\left(\frac{\mathrm{\Delta }R}{R}\right)}^2$

Calculate the percentage error in the measured value of g.

Calculate the percentage error in the measured value of g.

Deduce the value of g and its absolute uncertainty for this experiment.

Deduce the value of g and its absolute uncertainty for this experiment.

When d = 0.200 mm, s = 0.9 mm and D = 280 mm, determine the percentage uncertainty in the wavelength.

When d = 0.200 mm, s = 0.9 mm and D = 280 mm, determine the percentage uncertainty in the wavelength.

Determine the fractional uncertainty in v when T = 2.115 s, correct to one significant figure.

Determine the fractional uncertainty in v when T = 2.115 s, correct to one significant figure.

The lines of the minimum and maximum gradient are shown.

Estimate the absolute uncertainty in a.

The lines of the minimum and maximum gradient are shown.

Estimate the absolute uncertainty in a.

Suggest whether the data are consistent with the theoretical prediction.

Suggest whether the data are consistent with the theoretical prediction.

Show that the value of $\gamma$  is about 0.03.

Show that the value of $\gamma$  is about 0.03.

Identify the fundamental units of $\gamma$.

Identify the fundamental units of $\gamma$.

In order to find the uncertainty for $\gamma$, a maximum gradient line would be drawn. On the graph, sketch the maximum gradient line for the data.

In order to find the uncertainty for $\gamma$, a maximum gradient line would be drawn. On the graph, sketch the maximum gradient line for the data.

The percentage uncertainty for $\gamma$ is $15\%$. State $\gamma$, with its absolute uncertainty.

The percentage uncertainty for $\gamma$ is $15\%$. State $\gamma$, with its absolute uncertainty.

The expected value of $\gamma$ is $0.027$. Comment on your result.

The expected value of $\gamma$ is $0.027$. Comment on your result.

A student measures the length l and width w of a rectangular table top.

What is the absolute uncertainty of the perimeter of the table top?

A.  $0.3 \text{cm}$

B.  $0.6 \text{cm}$

C.  $\text{1.2} \text{cm}$

D.  $\text{2.4} \text{cm}$

A student measures the length l and width w of a rectangular table top.

What is the absolute uncertainty of the perimeter of the table top?

A.  $0.3 \text{cm}$

B.  $0.6 \text{cm}$

C.  $\text{1.2} \text{cm}$

D.  $\text{2.4} \text{cm}$

What is the unit of power expressed in fundamental SI units?

A.  ${\text{kg\hspace{0.05em}m\hspace{0.05em}s}}^{-3}$

B.  ${\text{kg\hspace{0.05em}m\hspace{0.05em}s}}^{-1}$

C.  ${\text{kg\hspace{0.05em}m}}^2{\text{\hspace{0.05em}s}}^{-1}$

D.  ${\text{kg\hspace{0.05em}m}}^2{\text{\hspace{0.05em}s}}^{-3}$

What is the unit of power expressed in fundamental SI units?

A.  ${\text{kg\hspace{0.05em}m\hspace{0.05em}s}}^{-3}$

B.  ${\text{kg\hspace{0.05em}m\hspace{0.05em}s}}^{-1}$

C.  ${\text{kg\hspace{0.05em}m}}^2{\text{\hspace{0.05em}s}}^{-1}$

D.  ${\text{kg\hspace{0.05em}m}}^2{\text{\hspace{0.05em}s}}^{-3}$

A ball of mass (50 ± 1) g is moving with a speed of (25 ± 1) m s⁻¹. What is the fractional uncertainty in the momentum of the ball?

A.  0.02

B.  0.04

C.  0.06

D.  0.08

A ball of mass (50 ± 1) g is moving with a speed of (25 ± 1) m s⁻¹. What is the fractional uncertainty in the momentum of the ball?

A.  0.02

B.  0.04

C.  0.06

D.  0.08

The intensity of a wave can be defined as the energy per unit area per unit time. What is the unit of intensity expressed in fundamental SI units?

A.  kg m⁻²s⁻¹

B.  kg m²s⁻³

C.  kg s⁻²

D.  kg s⁻³

The intensity of a wave can be defined as the energy per unit area per unit time. What is the unit of intensity expressed in fundamental SI units?

A.  kg m⁻²s⁻¹

B.  kg m²s⁻³

C.  kg s⁻²

D.  kg s⁻³

The radius of a circle is measured to be (10.0 ± 0.5) cm. What is the area of the circle?

A.  (314.2 ± 0.3) cm²

B.  (314 ± 1) cm²

C.  (314 ± 15) cm²

D.  (314 ± 31) cm²

The radius of a circle is measured to be (10.0 ± 0.5) cm. What is the area of the circle?

A.  (314.2 ± 0.3) cm²

B.  (314 ± 1) cm²

C.  (314 ± 15) cm²

D.  (314 ± 31) cm²

A rocket travels a distance of 3 km in 10 s.

What is the order of magnitude of $\frac{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{rocket}}{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{a} \mathrm{vacuum}}$?

A.  −5

B.  −6

C.  −7

D.  −8

A rocket travels a distance of 3 km in 10 s.

What is the order of magnitude of $\frac{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{rocket}}{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{a} \mathrm{vacuum}}$?

A.  −5

B.  −6

C.  −7

D.  −8

A rocket travels a distance of 3 km in 10 s.

What is the order of magnitude of $\frac{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{rocket}}{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{a} \mathrm{vacuum}}$?

A.  −5

B.  −6

C.  −7

D.  −8

A rocket travels a distance of 3 km in 10 s.

What is the order of magnitude of $\frac{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{rocket}}{\mathrm{the} \mathrm{speed} \mathrm{of} \mathrm{light} \mathrm{in} \mathrm{a} \mathrm{vacuum}}$?

A.  −5

B.  −6

C.  −7

D.  −8

Masses P and Q, are measured to be (30 ± 1) g and (20 ± 1) g respectively. Which of the following expressions gives the least percentage uncertainty?

A.  P + Q

B.  P − Q

C.  P × Q

D.  $\frac{P}{Q}$

Masses P and Q, are measured to be (30 ± 1) g and (20 ± 1) g respectively. Which of the following expressions gives the least percentage uncertainty?

A.  P + Q

B.  P − Q

C.  P × Q

D.  $\frac{P}{Q}$

Masses P and Q, are measured to be (30 ± 1) g and (20 ± 1) g respectively. Which of the following expressions gives the least percentage uncertainty?

A.  P + Q

B.  P − Q

C.  P × Q

D.  $\frac{P}{Q}$

Masses P and Q, are measured to be (30 ± 1) g and (20 ± 1) g respectively. Which of the following expressions gives the least percentage uncertainty?

A.  P + Q

B.  P − Q

C.  P × Q

D.  $\frac{P}{Q}$

Calculate, in nm, $\lambda$.

Calculate, in nm, $\lambda$.

Calculate, in nm, $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

The student moves the screen closer to the double slit and repeats the measurements. The instruments used to make the measurements are unchanged.

Discuss the effect this movement has on the fractional uncertainty in the value of $\lambda$.

Estimate, using the graph, the maximum height of the bottle.

Estimate, using the graph, the maximum height of the bottle.

Estimate, using the graph, the maximum height of the bottle.

Estimate the acceleration of the bottle when it is at its maximum height.

Estimate the acceleration of the bottle when it is at its maximum height.

Estimate the acceleration of the bottle when it is at its maximum height.

Draw and label on diagram B the forces acting on the sphere just after it has been released.

Draw and label on diagram B the forces acting on the sphere just after it has been released.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

Deduce the unit of μ in terms of fundamental SI units.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

The percentage uncertainty of the gradient is 6.0 %. The frequency $f$ of the wave is (60.0 ± 2.0 %) Hz.

Estimate, using the answer to (b)(iv), μ for the string. Include the percentage uncertainty of μ in your answer.

State the unit for k.

State the unit for k.

State the unit for k.

State the unit for k.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Calculate the absolute uncertainty in $F_{\max }^3$ for Δp = 30 kPa. State an appropriate number of significant figures for your answer.

Show, using two suitable data points from the table, that the student’s initial hypothesis is not supported.

Show, using two suitable data points from the table, that the student’s initial hypothesis is not supported.

Show, using two suitable data points from the table, that the student’s initial hypothesis is not supported.

Show, using two suitable data points from the table, that the student’s initial hypothesis is not supported.

Draw the absolute uncertainty determined in part (d)(i) as an error bar on the graph.

Draw the absolute uncertainty determined in part (d)(i) as an error bar on the graph.

Draw the absolute uncertainty determined in part (d)(i) as an error bar on the graph.

Draw the absolute uncertainty determined in part (d)(i) as an error bar on the graph.

Draw on the graph the position of the missing point for the Δp value of 40 kPa.

Draw on the graph the position of the missing point for the Δp value of 40 kPa.

Draw on the graph the position of the missing point for the Δp value of 40 kPa.

Draw on the graph the position of the missing point for the Δp value of 40 kPa.