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Activity 5a Gravity and Mass of the Earth Objectives: The purpose of this lab is to measure g, the acceleration of gravity at the surface of earth, and use it to calculate the mass of the Earth. Introduction: The Gravitational Constant The force felt on an object on earth is due to gravity which is defined with the relations: F_g = mg (1) Where Fg is the force due to gravity and m is the mass of the object and g is the acceleration due to gravity that is felt on the Earth's surface. The value of g depends on the mass of the Earth and the distance from the center of the Earth, as well as the universal gravitational constant, G. Thus the force felt on an object due to gravity can also be defined as: F_g=GMm/R^2 (2) Where M is the mass of the Earth, m is the mass of the object, and R is the distance to the center of the Earth to the center of the object. Activity I: Mathematical activity Put the two equations together and write g in terms of G, M, and R. Does the acceleration due to gravity, g, depend on the mass of the object being accelerated? Acceleration of a Dropped Object When an object is dropped from a height h above the surface of the earth, the amount fallen by the object is time dependent and it is given by the relation: h=1/2 gt^2 (3) The new term t in equation (3) is the time it takes for the object to drop. Equipment: A golf ball (or similarly heavy, small object); A stopwatch (can be a phone); Height of at least 4-5 meters (about 15ft); A string and ruler, or other method of measuring height Activity II: Dropping Objects With equation (3) g can be deduced if h and t are known. Thus we will measure both the height and time it takes to drop in other to deduce g. First, select a place to drop your ball, and measure the height. One method is to use a piece of string long enough to reach from the top to the bottom, and then measure the length of the string. This may be easiest with the help of a friend to hold the other end of the string or tape measurer. Make sure to use a metric measuring device (meters). If you only have a way to measure feet you will have to convert to meters (1ft = .305m) What is the height you will drop from? (in meters) We also need to characterize our measurement error/uncertainty. You should think about whether or not you stretched your string at any point, how big your ruler was compared to the string, and how finely you could measure the string compared to the ruler. With this in mind, estimate how far off your measured height could be from the actual height in meters. This is called the uncertainty in height. You can calculate the percent uncertainty by dividing the uncertainty in height by the total height measurement. What is your percent uncertainty for the height measurement? Time the Drop. Do not record the first two drops as you get used to the setup. Repeat 20 more times in order to make sure you've timed the drop of the ball well. This minimizes some of the errors that have to do with stopping and starting the stopwatch at the right time. Be careful to drop the ball from the exact height you measured, and to drop the ball rather than throw it. Tabulate your results below Drop # Time/(s) Drop # Time/(s) Drop # Time/(s) 1 8 15 2 9 16 3 10 17 4 11 18 5 12 19 6 13 20 7 14 Find the average time of all the 20 drop times in the table above. Estimate the error ∆t in the time measurements using the relations. ∆t=(t_high-t_low)/2 (4) Rearrange equation (3) and write g in terms of h and t. Use your measured values for h and t to calculate g. It is essential to note that g has been reported precisely at g = 9.8m/s2. Calculate the difference (subtraction) between the reported value of g and the value you calculated in the previous step. The difference between the two numbers (the reported constant and your calculated value) is due to errors in the measurement of h and t. In your estimation, which error had a larger effect on the result, Δh or Δt (hint: compare their percent uncertainties)? Next, we are going to use the errors in the measurement of h and t to calculate the range of possible values of g. Let’s define: t_max=t+∆t t_min=t-∆t h_max=h+∆h h_min=h-∆h Such that the upper value of g will be calculated with h_max and t_min and lower values of g will be calculated with h_min and t_max with the relations; g_max=(2h_max)/(t_min^2 ) and g_min=(2h_min)/(t_max^2 ) What are your upper and lower values for g? Is the real value of g between your calculated g_min and g_max? What does it mean if the real value of g is not within your range of possible g? Part III: Mass of the Earth For this section we will use equations (1) and (2). In order to find the mass of the earth, the universal constant G and the radius of the earth R are needed. The known values are: G=6.67×10^(-11) m^3⁄(kgs^2 ) R=6.40×10^6 m The reason we are giving you the value for R and using it to calculate M (instead of the other way around) is because R is much easier to measure than M. The radius of the Earth can be measured with a few observations and some geometry. The mass is much harder to measure directly. In the earlier section we wrote g in terms of M, R, and G. In this section you should put equation (1) and (2) together and find M in terms of g, G and R What equation did you come up with? Calculate a value for M using the g value you obtained earlier (in the part above). Use your g_min and g_max to calculate a Mmin and Mmax. Calculate the real value of M using the real value of g = 9.8m/s^2. How does it compare to your value of M (calculate the difference)? Is the real value of M between your Mmin and Mmax? Do you think this experiment is a reliable way to calculate the mass of the Earth? Explain. Part IV: Sources of Error List 3 things that made your measurement of g more uncertain. Of the three sources of error you have listed, pick one and explain what changes you could make to reduce or eliminate this error/uncertainty.

Answer 1

Rest of the part is about doing and experiment and getting real life results. I hope you would do that part.

Thanks and let me know if any doubt regarding my answer.