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Gravity
Learning Goals for this Lab:
Measure the gravitational force originating from a 1.5 kg lead sphere.
Determine G, the universal gravitational constant.
Derive an expression for the gravitational constant in terms of known values bycombining concepts from many topics in the course including kinematics, Newton’s 2ndLaw, torque, equilibrium, restoring forces, and gravitation.
Newton’s law of universal gravitation states that the force acting between two masses 𝑀 and 𝑚 separated by a distance 𝑟 is: 𝐹=𝐺𝑀𝑚𝑟2
where 𝐺 is the universal gravitational constant (𝐺=6.67 x 10-11 Nm2/kg2). This law is very useful for things like searching for extrasolar planets by the small gravitational effects they have on their sun, and for determining the path of spacecraft traversing the solar system, such as the New Horizons spacecraft that recently completed a 9-year journey to Pluto. Newton’s law of universal gravitation holds for most commonly encountered masses and separation distances, but can be considered a special case of Einstein’s theory of general relativity, the latter of which is accurate for very large masses and very close distances.
Pre-Lab
In this experiment we will determine the gravitational pull of heavy lead sphere by measuring the acceleration of a small sphere towards the large sphere. Which of the following causes the acceleration?
a)The gravitational force originating from the large sphere.
b)The torsion in the wire from which the small sphere is suspended.
c)The gravitational force originating from the earth.
d)a) and b)
In this experiment, we calculate the gravitational force originating from a 1.5 kg lead sphere. To carry out the experiment, two large spheres are placed close to small spheres suspended from a fine torsion wire. The gravitational force exerted by the large spheres attracts the small spheres, rotating them closer and producing torsion in the wire (Figure 1, Left Panel). When the balance comes to equilibrium, the gravitational force of the large spheres is balanced by the torsion in the wire. Then, if the large spheres are moved to the opposite position, see Figure 1, Right Panel, their gravitational force works with the restoring force from the wire, and the small spheres accelerate back toward their original position. The
Gravity
initial acceleration is essentially constant, and it is measured and used to find the gravitational force originating from the large spheres. For this reason, this method is called the initial acceleration method.
Figure 1.
Cavendish balance, laser, stopwatch, measuring tape, meterstick
Part I.
Data Collection
Read through these instructions fully before continuing. The balance requires several hours to come to equilibrium, so there is only time for one data collection run per class period.
Do not touch the balance until instructed. The balance is has been fine-tuned and brought to equilibrium and is very sensitive to vibration. Take special care not to bump the table. During data collection, minimize all sources of vibration such as stomping on the floor or moving chairs.
For data collection, we will simply rotate the large spheres all the way around to their extreme opposite position and, as the small spheres accelerate towards the large spheres, we will mark the position of the laser dot as a function of time.
1.The Cavendish balance has already been centered for you. The large spheres should be in their fullyrotated position and the small spheres should already be in equilibrium (see Fig. 1); check with yourinstructor to make sure this is the case.
2.Without touching anything, visually inspect the apparatus and locate the large spheres, smallspheres, torsion wire, and mirror.
3.After you rotate the large spheres, the laser dot on the screen will accelerate back towards thecenter and you must be prepared to record the position of the dot every 15 seconds. The distancewill be only a few millimeters per 15 seconds at first, and then will increase. Assign one groupmember to be responsible for rotating the spheres, one to mark the position of the dot, and one tobe the timer calling out the 15 second marks. Practice a dry run of the experiment, as you will onlyhave time for one data collection run.
4.Prepare the stopwatch and a pencil to mark the position of the laser dot. Mark the initial position ofthe dot.
5.Gently rotate the two large lead balls all the way to the opposite orientation, immediately start thetimer, and record the position of the laser dot every 15 seconds for two minutes.
Part II.
Part II. AnalysisAnalysis
We derive the expressions required to calculate the gravitational force and constant G from the measured values. We can restrict our analysis to one of the small spheres because of the symmetry of the apparatus.
1.Draw a free-body diagram indicating the two forces acting on a small sphere while it is inequilibrium (i.e. before we shift the larges spheres to their new position).
2.Draw a second free body diagram for the forces acting on the small sphere for the instant just afterthe large spheres are switched to their new position.
3.Looking at your free body diagrams, you can see that once we switch the large sphere to its newposition, the gravitational force will be acting in the same direction as the restoring force from thewire. Newton’s 2nd Law states that the sum of the forces causes its acceleration. Write down theequation of motion (i.e. the specific version of Newton’s 2nd Law) for the instant the small spherebegins accelerating just after we switch the large spheres to their new position. Refine your equationto ultimately be in terms of 𝐺, 𝑀, 𝑚, 𝑟, and 𝑎 (where 𝑎 is the acceleration of the small sphere).
4.Rearrange your equation for 𝐺. Now we see that we only need to find 𝑎, 𝑀, and 𝑟 to calculate 𝐺.
Measure 𝑀 using a balance, taking care not to drop the heavy sphere.
Ignoring the miniscule distance the small sphere moves, 𝑟 is simply the half the thickness of theapparatus (0.0146 m) plus the radius of the large sphere (0.0318 m). Thus, 𝑟 = 0.0464 m.
5.Now we only need the acceleration, which is why we measured the distance as a function of time ofthe laser dot. But we must first convert from the dot’s motion to the sphere’s motion.
Derive an expression for the acceleration of the sphere 𝑎 as a function of: the acceleration of thedot 𝑎dot, the distance of the screen from the mirror 𝐿, and the sphere’s radius of rotation 𝑑.
Assume that this acceleration is constant for a short time period after the sphere startsmoving. (Which equations are applicable when acceleration is constant?) We will seeshortly that the constant acceleration approximation is quite valid.
Making a sketch may be helpful.
Plug this equation for 𝑎 into the equation for 𝐺 you derived above. We now have an equation for 𝐺 in terms of measurable quantities.
6. Plot your data of position of the dot 𝑥dot vs. 𝑡2. Is it initially linear? Was the approximation of constant acceleration valid? Reduce the data range to cover the linear region from approximately 15 seconds to 90 seconds. Fit the data to a line and extract the slope. What does the slope represent in this plot?
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7. Use the value of the slope along with your derived equation to calculate the gravitational constant 𝐺. You will need to measure 𝐿. 𝑑 = 0.05 m for this apparatus.
8. To this point, we have ignored the gravitational force from the second large sphere. Even though it is farther away, it still exerts a non-negligible force on our small sphere. The force from this second sphere 𝑓 can be expressed in terms of the force from the nearer sphere 𝐹 using a correction factor 𝛽 as in 𝑓=𝛽𝐹.
Show that 𝛽=𝑟3(𝑟2+4𝑑2)32⁄
9. Calculate the percent error between the known value for the gravitational constant and your calculated value of 𝐺′ where 𝐺′=(1+𝛽)𝐺.
10. Propose another way that this apparatus could be used to measure the gravitational force.
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Part I
Part IIII. Results and DiscussionI. Results and Discussion
In your own words, summarize your major results. For example, did your experimental results corroborate the theoretical predictions? Were your results expected? Unexpected? Why? Support your statements by referring to your data, graphs, observations, calculations, and possible sources of error. Be sure to upload your Excel files and any additional files you create such as figures or sketches. Refer to the rubric for the detailed criteria on which your grade will be based

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