### The Idea

This classic experiment explores the connection between an object's acceleration and the force applied to it. This fundamental principle of physics was first formulated by Sir Isaac Newton in the famous second law of motion that bears his name. To measure acceleration, you use either the stopwatch or the motion sensor technique of measuring acceleration, which we used in previous experiments. The force will be provided courtesy of the Earth, in the form of the gravitation force on a mass hanging from a string.

### What You Need

- low-friction cart (or an air track and glider, if available)
- spring scale
- mass set (including 50 g, 100 g, 200 g)
- tape
- string
- pulley (low mass and low friction is preferable)
- clamp to attach the pulley to the table
- table top (at least 1 meter in length)
- stopwatch and meterstick or motion sensor

### Method

- Determine the mass of the cart in grams. Divide by 1000 to get kilograms.
- Place a 100g (0.1kg) mass in the cart. Secure it with tape, if necessary.
- Set the cart at one end of the table, and attach the pulley to the other end.
- Attach the string to the cart, run it over the pulley, and tie a loop that extends a few inches below the edge of the table, in the other end, as shown in Figure 25-1.
- While holding the cart in position at the far end of the table, hang a mass on the loop on the other side of the string.
- Next, you release the cart and let the weight of the hanging mass pull the cart across the table. As you do this, you measure the acceleration of the cart using either of the previous methods:
- Stopwatch: measure the time (in seconds) for the cart to be pulled a measured distance (in meters). The acceleration (in m/s
^{2}) is determined by a = 2d/t^{2}, where*d*is the distance that the object is pulled across the table (in m) during time,*t*(in seconds). - Motion sensor: record the position of the cart as it is drawn across the table. Display the velocity versus time graph and determine the acceleration of the cart by finding the slope of that graph. This can be done either using the Slope tool from the DataStudio menu or more simply by obtaining the acceleration as the change in velocity divided by the change in time.

- Stopwatch: measure the time (in seconds) for the cart to be pulled a measured distance (in meters). The acceleration (in m/s
- Repeat this measurement, but make the following changes:
- Vary the mass in the cart, but keep the applied force constant, as indicated in Figures 25-2 and 25-3.
- Vary the applied force by adding or removing some of the
*hanging weight*, but keep the mass in the cart constant, as shown in Figure 25-4.

### Proving Newton's second law

Newton's second law, which states that F = ma, or as Newton originally put it, a = F/m.

*m represents the entire mass of the system and includes the mass of the cart (m*._{c}), plus the mass in the cart (m_{1}) plus the hanging mass (m_{2})*F*is the applied force that pulls the cart and is given by the hanging mass, m_{2}(in*kilograms*, not in grams) times the gravitational acceleration (9.8 m/s^{2}). (To get kilograms from grams, divide the number of grams by 1000.)*a*is the acceleration (in m/s^{2}) of the*entire system*, including the cart, its contents, and the hanging mass.

You can use the following to organize your data:

### Expected Results

### The effect of force on acceleration

For a fixed mass in the cart (m_{1}), the greater the applied force, the greater the acceleration. The expected relationship between acceleration and force is shown in Figure 25-4, which (for simplicity) shows the effect of increasing the hanging mass on acceleration. (The actual driving force is given in newtons, which is simply 9.8 times the mass in kilograms.) Without friction (and to the extent that friction is eliminated from this experiment), this should be a linear relationship as indicated in Figure 25-5.

### The effect of mass on acceleration

For a given applied force, the heavier the load, the smaller the rate of acceleration. This is an *inverse* relationship, as shown in Figure 25-6.

Experimental results for acceleration for a given mass and applied force come close to the predicted results if the frictional forces are not significant. Even with friction, it can still be shown that acceleration depends on applied force and is inversely proportional to the mass. Friction increases when too much mass is placed in the cart. However, if the mass is too small, the acceleration can be so high, it becomes more difficult to measure accurately.

Use of low-friction tracks reduces the amount of friction. Motion sensors provide a nice way to determine the acceleration. Figure 25-7 shows the result of motion sensor data for two different total accelerated masses.

### Why It Works

Newton's second law states that F = ma or a = F/m. More force leads to greater acceleration, but more mass lowers the rate of acceleration.

### Other Things to Try

You may want to consider doing this using a Hover Puck drawn across the floor by a mass hung from a pulley, as shown in Figure 25-8. As before, remember to include the mass of the Hover Puck as part of the total system mass being accelerated. The higher the pulley is supported above the floor, the longer the run you can have across the floor. A qualitative but very intuitive way of showing the relationship between a force and acceleration can be shown using an LED accelerometer. The constant force from the fan results in an acceleration indicated by the LEDs as shown in Figure 25.9. The direction of the force vector is in the same direction as the acceleration vector.

### The Point

The result here is one of the most significant results in physics: a force causes acceleration. For a given mass, the acceleration of an object is proportional to the applied force. For a given force, the acceleration is inversely proportional to the amount of mass.

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