# Measuring g

## The Acceleration Due To Gravity

#### Purpose

To determine the value of the acceleration due to gravity

#### Objectives

• To determine the acceleration of gravity by measuring the acceleration of a system of two objects and plugging it into a theoretical formula.
• To learn how to compare the results from different runs for which you measure an experimental quantity.
• To learn how to estimate the accuracy of an experimental result by comparing it to an established value.

#### Equipment

• Air track
• Photogate connected to PASCO Interface
• Glider with picket fence
• A pulley, hook, hanging weights
• Mass Scale

#### Theory The acceleration of a system of objects is always determined by Newton’s Second Law:

Net Force = (Mass of the system) × (The acceleration of the system)

In this experiment, we will consider a system of two objects connected with a string to each other. The first object will be a glider with a picket fence that will move along a frictionless horizontal track. The second object will be a hook and weights hanging vertically. The two objects will be connected by a string that goes over a frictionless pulley, as shown on the diagram above.

According to Newton’s Second Law, the acceleration of such a system will be determined by the net external force applied on the system as a whole. Here, we neglect those internal forces that hold the objects together, as they cannot change the overall acceleration of the system. Also, as long as the string is tout, both objects will be moving as “one”, that is they will have equal acceleration that will be the acceleration of the system.

In this particular case, when the track is horizontal and in the absence of friction, the force responsible for the acceleration of the object is the gravitational pull on the hanging weights, that is

F = (mass of hanging object) × (acceleration due to gravity).

On the other hand, that force F will cause both objects to begin moving, thus it must be equal to :

F = (mass of the two objects) × (acceleration of the two objects)

When we combine the two expressions, we get the following equation:

(mass of hanging object) × (acceleration due to gravity) = (mass of the two objects) × (acceleration of the two objects)

In symbols, the formula is written as:

mhanging × g = (msliding  +  mhanging) × a

The gravitational acceleration can be calculated, then, from the following formula:

g = [(msliding  +  mhanging) × a] ⁄ [mhanging ]

Note. The mass of the hanging objects includes everything hanging, that is hook and weights.

Note. The mass of the sliding object includes everything that slides along the ramp, that is glider, picket fence, any pieces attached to the glider, etc.

#### Experiment, Data and Results

##### Preliminary Setup
• Level the air track. Place the glider with the picket fence on the air track while the air pump is on. It should not drift in any direction. If it drifts, adjust the height of the supports.

• Attach objects. Put a hooked end piece into the front end of the glider. Attach the string to that hook. Clamp a pulley to the table edge and put the string over it. Attach the hook to the loose end of the string. Attach a 2-g cylinder to the hanging hook.
• Photogate and Computer Set up. Follow the Picket Fence and Photogate Instructions to set up the computer.

• Photogate Check. As the glider with the picket fence glides along the ramp, the photogate must blink exactly once for each time when a darkened space goes through it.

#### Activity

We will measure the acceleration of the glider, a. That acceleration is equal to the acceleration with which the hanging object is falling, and to the acceleration of the entire system.

1. Measure the mass of the gliding object, msliding. Do not forget to include everything: glider,picket fence, and anything attached to them!
2. Measure the hanging mass, mhanging, including the hook and the cylinder.
3. Place the glider on the track next to the photogate, so that when it begins to move it goes through the photogate.
4. Turn on the Air Pump and click on Start the experiment.
5. Let go of the glider. The photogate will measure how fast each of its darkened segements go through it, and will send that information to the computer, where the computer will perform the calculations and will give the measured acceleration of the glider.
6. Stop the experiment once the glider has passed through the photogate and before the hanging weights have hit the ground.
7. Record the acceleration, aglider, that you read on the computer screen.
8. Run 2. Add noter 2.0-g cylinder (or if simply replace the old 2.0-g with a new 4.0-g cylinder) on the hook and re-measure the hanging object mass. Repeat the experiment and record the acceleration.
9. Run 3. Keeping everything as before in Run.2, add a 50-g cylinder on each side of the glider and re-measure the sliding object mass, mslidng. Repeat the experiment.

For the three runs, write down your results in the following table.

 Hanging Mass [g] Sliding Mass [g] Total Mass [g] Glider Acceleration [m/s2] Gravitational Acceleration [m/s2] mhang mslid mtotal a [m/s2] g [m/s2]

#### Questions.

Agree or Disagree? Compare the value for the acceleration due to gravity for all three runs. Look at their leading digits and see if they are the same. If all your three results begin with the same first two digits, we will say that the results agree very well. If the three results begin with the same first digit but disagree in the second, we will say that the results agree somewhat. And, if the three results begin with different first digit, we will say that there is a disagreement.

Following the above guidelines, how do your results agree

Agree Well
Somewhat Agree
Disagree

If they disagree, you must

1. Identify which run is the most suspicious and repeat it, and
2. List the factors that might have contributed to the “Disagree”-ing results:
Using the results from the three runs, what is the average value for the acceleration due to gravity?

Is it correct? In order to find out whether your results are correct, you must first find what is the “correct” value for the acceleration of gravity at your location.

You can find the theoretical value of “g” by going to Wolfram Alpha and typing “gravitational acceleration washington dc”. Write down the theoretical value for g.

To how many digits do your result and the “g” agree?