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Complete the table and plot a graph of T2 against l for a simple pendulum experiment.

Physics

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2 parts
b

3.928 s2/m\text{3.928 s}^2/\text{m}

c

10.05 m/s2\text{10.05 m/s}^2

Step-by-step solution

3 steps

The image shows an experiment involving a simple pendulum. The table provides data for the length of the pendulum (ll) and the time taken for 20 swings. The period TT and T2T^2 values are also partially filled.

First, let's complete the table for T(s)T(s) and T2(s2)T^2(s^2).
The period TT is the time for one swing. Since the mean time is for 20 swings, T=Mean t20T = \frac{\text{Mean t}}{20}.
Then, T2T^2 is the square of the period.

Here's the completed table:

ll (cm)ll (m)t1t_1 (s)t2t_2 (s)Mean tt (s)TT (s) = Mean tt/20T2T^2 (s2^2)
1001.0040.3340.0640.192.00954.038
900.9038.0038.1038.051.90253.620
800.8036.0036.1036.051.80253.249
700.7034.0033.9033.951.69752.881
600.6031.0031.0031.001.55002.403
500.5029.0028.6028.801.44002.074

Part a

Plot a graph T2(s2)T^2(s^2) against ll in meters.

This step requires plotting on graph paper, which cannot be done here. However, I can provide the data points for plotting.
The points to plot are (l,T2)(l, T^2):
(1.00,4.038)(1.00, 4.038)
(0.90,3.620)(0.90, 3.620)
(0.80,3.249)(0.80, 3.249)
(0.70,2.881)(0.70, 2.881)
(0.60,2.403)(0.60, 2.403)
(0.50,2.074)(0.50, 2.074)

Part b

3.928 s2/m\text{3.928 s}^2/\text{m}

Find the slope of the graph.

The formula for the period of a simple pendulum is T=2πlgT = 2\pi\sqrt{\frac{l}{g}}.
Squaring both sides gives T2=4π2lgT^2 = 4\pi^2 \frac{l}{g}.
This equation is in the form y=mxy = mx, where y=T2y = T^2, x=lx = l, and the slope m=4π2gm = \frac{4\pi^2}{g}.

To find the slope, we can use two points from the calculated data. Let's use the first and last points for better accuracy.
Point 1: (l1,T12)=(1.00 m,4.038 s2)(l_1, T_1^2) = (1.00 \text{ m}, 4.038 \text{ s}^2)
Point 2: (l2,T22)=(0.50 m,2.074 s2)(l_2, T_2^2) = (0.50 \text{ m}, 2.074 \text{ s}^2)

  1. Calculate the slope mm.

    m=T12−T22l1−l2m = \frac{T_1^2 - T_2^2}{l_1 - l_2} m=4.038 s2−2.074 s21.00 m−0.50 mm = \frac{4.038 \text{ s}^2 - 2.074 \text{ s}^2}{1.00 \text{ m} - 0.50 \text{ m}} m=1.964 s20.50 mm = \frac{1.964 \text{ s}^2}{0.50 \text{ m}} m=3.928 s2/mm = 3.928 \text{ s}^2/\text{m}

    The slope of the graph is 3.928 s2/m\boxed{\text{3.928 s}^2/\text{m}}.

Part c

10.05 m/s2\text{10.05 m/s}^2

Calculate the value of gg.

From the relationship m=4π2gm = \frac{4\pi^2}{g}, we can find gg.

  1. Rearrange the formula to solve for gg.

    g=4π2mg = \frac{4\pi^2}{m}
  2. Substitute the value of the slope mm.

    g=4×(3.14159)23.928 s2/mg = \frac{4 \times (3.14159)^2}{3.928 \text{ s}^2/\text{m}} g=4×9.86963.928 s2/mg = \frac{4 \times 9.8696}{3.928 \text{ s}^2/\text{m}} g=39.47843.928 s2/mg = \frac{39.4784}{3.928 \text{ s}^2/\text{m}} g=10.050 m/s2g = 10.050 \text{ m/s}^2

    The value of gg is 10.05 m/s2\boxed{\text{10.05 m/s}^2}.

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