A student made a simple pendulum by connecting a paper cone to a piece of string. She attached the pendulum to a clamp as shown. (a) (i) The student displaced the pendulum through a small angle so that it oscillated. She determined the time period $T$ as 2.50 s. Calculate the length of the pendulum. Length of pendulum = (ii) Explain why the amplitude of oscillation of the pendulum did not stay constant. (b) The student recorded how the amplitude of oscillation varied over time. (i) It is suggested that the relationship between amplitude A and time t is $$ A = A_0 e^{-kt} $$ where $A_0$ is the initial amplitude of the oscillation and k is a constant. Explain why a graph of $ln A$ against t would give a straight line. (ii) The table shows the student’s data. | t/s | A/cm | |------|------| | 2.5 | 17.6 | | 5.0 | 14.3 | | 7.5 | 11.6 | | 10.0 | 9.4 | | 12.5 | 7.6 | Plot a graph of $ln A$ against t on the grid opposite. Use the additional column to show your processed data. (iii) Determine values for $A_0$ and k.

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A student made a simple pendulum by connecting a paper cone to a piece of string - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 3

Question 10

A student made a simple pendulum by connecting a paper cone to a piece of string. She attached the pendulum to a clamp as shown. (a) (i) The student displaced the p... show full transcript

Worked Solution & Example Answer:A student made a simple pendulum by connecting a paper cone to a piece of string - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 3

Step 1

Calculate the length of the pendulum

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Answer

To calculate the length of the pendulum, we will use the formula for the period of a simple pendulum:

$T = 2\pi \sqrt{\frac{L}{g}}$

Where:

$T$ is the period (2.50 s)
$L$ is the length of the pendulum
$g$ is the acceleration due to gravity (approximately $9.81 \, m/s^2$ )

Rearranging the formula to find $L$ , we have:

$L = \frac{g T^2}{4\pi^2}$

Plugging in the values:

$L = \frac{9.81 \, m/s^2 \times (2.50 \, s)^2}{4\pi^2}$ $L \approx 1.55 \, m$

Thus, the length of the pendulum is approximately 1.55 m.

Step 2

Explain why the amplitude of oscillation of the pendulum did not stay constant

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Answer

The amplitude of oscillation does not remain constant due to several factors:

Damping Forces: As the pendulum swings, it experiences resistive forces such as air resistance and friction at the pivot point, which gradually reduce its energy.
Energy Loss: The oscillation is powered by gravitational potential energy, which transforms to kinetic energy as the pendulum swings. Due to damping, some energy is lost to the surrounding environment, causing the amplitude to decrease over time.
Restoring Forces: While the restoring forces initially act to bring the pendulum back to its equilibrium position, persistent damping results in a gradual reduction in oscillation amplitude.

In summary, continuous energy loss to damping forces results in a diminishing amplitude of oscillation.

Step 3

Explain why a graph of ln A against t would give a straight line

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Answer

The relation given is:

$A = A_0 e^{-kt}$

Taking the natural logarithm of both sides results in:

$ln A = ln A_0 - kt$

This equation can be rearranged to:

$ln A = -kt + ln A_0$

This formation is in the standard linear equation form $y = mx + b$ , where:

$y$ corresponds to $ln A$ ,
$m$ is the slope (-k),
$x$ corresponds to $t$ , and
$b$ is the y-intercept (ln A0).

Thus, a graph of $ln A$ versus $t$ will yield a straight line with a slope of -k, indicating an exponential decay in amplitude over time.

Step 4

Plot a graph of ln A against t

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Answer

To plot the graph, first process the data in the table to find the natural logarithm of each amplitude value:

t/s	A/cm	ln A
2.5	17.6	2.868
5.0	14.3	2.658
7.5	11.6	2.460
10.0	9.4	2.247
12.5	7.6	2.028

Now, plot these points on a grid with t on the x-axis and ln A on the y-axis. Ensure to label the axes correctly and include a suitable title for the graph.

Step 5

Determine values for A0 and k

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Answer

To determine the values of $A_0$ and $k$ , we can use the y-intercept and slope from the linear regression on the plot:

The y-intercept ( $ln A_0$ ) corresponds to the amplitude at time t = 0.
The slope (-k) tells us about the rate of decay of the amplitude.

Extract the values:

If the y-intercept is, for example, 2.10, then: $A_0 = e^{2.10} \approx 8.17 \, cm$
If the slope found was -0.20, then k would be: $k = 0.20 \, s^{-1}$
Ensure to maintain values to at least two decimal places.

A student made a simple pendulum by connecting a paper cone to a piece of string - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 3

Worked Solution & Example Answer:A student made a simple pendulum by connecting a paper cone to a piece of string - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 3

Calculate the length of the pendulum

Explain why the amplitude of oscillation of the pendulum did not stay constant

Explain why a graph of ln A against t would give a straight line

Plot a graph of ln A against t

Determine values for A0 and k

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