Consult Conceptual Example 9 in preparation for this problem. Interactive LearningWare 6.3 also provides useful background. The

The formula for range is:

Given values are:

where θ equals 14.1 degrees

Using the equation above,

The calculated range is 66.7 meters.

Therefore, the range is approximately 66.1 meters.

assuming north-south is along the Y-axis and east-west along the X-axis

X = total X-displacement

from the graph, total displacement in the X-direction is computed as

X = 0 - 20 + 60 Cos45 + 0

X = 42.42 - 20

X = 22.42 m

Y = total Y-displacement

from the graph, total displacement in the Y-direction is computed as

Y = 40 + 0 + 60 Sin45 + 50

Y = 90 + 42.42

Y = 132.42 m

To calculate the magnitude of the net displacement vector, we apply the Pythagorean theorem, yielding

magnitude: Sqrt(X² + Y²) = Sqrt(22.42² + 132.42²) = 134.31 m

Direction: tan⁻¹(Y/X) = tan⁻¹(132.42/22.42) = 80.4 deg north of east

Answer:

1)  g = 4π² / m, 3) on the x-axis we have the pendulum lengths, while the y-axis shows the squared periods.

Explanation:

a) learners can model this system as a simple pendulum, where the angular velocity is given by

         w = √ g / l

Here, angular velocity, frequency, and period are interconnected:

         w = 2π f = 2π / T

Substituting yields:

         T = 2π√ l / g

Using this formula, students can calculate the gravitational acceleration by measuring the period for several pendulum lengths and plotting:

        T² = 4π²  l / g

We plot T² against l.

This represents a linear equation where T² is on the y-axis and l is on the x-axis:

        y = (4π² / g) l

The slope is given by:

         m = 4π² / g

Solving for g gives:

         g = 4π² / m

The slope is determined from the line's values rather than experimental data.

2) To perform the experiment, the string is secured to the sphere, then the pendulum length from the pivot to the sphere's center is measured using a tape measure. A slight angle (less than 10 degrees) is released, allowing the first swing to occur. Generally, the time for several oscillations, usually 10 or 20, is tracked to find the period:

    T = t / n

Next, a table is created comparing T² to the length, plotted with length on the x-axis to find the slope, from which the gravitational acceleration is derived.

3) The independent variable, which is the length of the pendulums, is plotted on the x-axis, while the dependent variable, the squared period, is on the y-axis.

4) Referring to the line equation:

            m = 4π² / g

             resulting in:

            g = 4π² / m

5) Once the spring is cut, the sphere continues to be influenced by gravitational acceleration. The harmonic motion ceases, and the sphere moves vertically.