# Discovering Newton's 2nd Law

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## Objective

Using a cart and pulley, students will determine the relationship between net force and acceleration for a fixed mass .

#### Big Idea

F=ma can be determined experimentally.

## Context and Supplies

Students have had a lot of practice determining net force and applied that to the concept of accelerated motion.  Now it is time to determine the mathematical relationship between the force acting on a mass and the masses acceleration.  Students will experimentally determine Newton's 2nd Law  (F=ma) where a cart is accelerated by a mass that is hanging over a pulley at the end of the track.  It is a great activity where my students a plan and conduct an experiment and collect data that they graph and analyze.

Since students will be engaged in an experiment seeking to figure out the relationship between the net force on an object and its resulting acceleration, students apply the concepts within Science Practice 3: Planning and carrying out investigations and Math Practice 2: Reason abstractly and quantitatively.  Ultimately, the students will determine that F=ma which is NGSS Performance Standard HS-PS2-1, the mathematical relationship between the net force acting on an object and its acceleration.

At each of the six lab tables, students set up the following supplies: a 1-meter track and cart, meter stick, stop watch, pulley that clamps to the track, string and masses.  0.025 kg is a good starting mass with each data point adding another 25 gram mass.  We don't have mass sets in my classroom, so I substitute known masses with large 1.5 inch washers that were purchased at a hardware store.  Each one is between 20-30 grams, which is just right for this experiment.  Students can start with one washer and add a washer for each data point.  Since the mass of each washer is not fixed, students are warned to measure the mass used for each data point.

## Laboratory Introduction

5 minutes

The first slide from the F=ma experiment Power Point is displayed as students come into the classroom.

When class begins, I move the to second slide and tell the students that they are experts at determining net force and are able to identify if an object is accelerating or not based on the net force. But I enthusiastically tell students, that is not enough! We cannot make any real predictions about an objects motion if we don't know the value of the acceleration or mass of that object.  The third slide shows today's goal, which is to "Determine the mathematical relationship between the net force on an object and the acceleration it experiences".

With a lab station set up at the front of the classroom, I demonstrate how the cart is accelerated by a hanging mass. If I add more mass to the hanging mass, students can see that the cart has a bigger acceleration. Students choose a hanging mass amount and measure the acceleration of the cart by using time, distance and the kinematic equation x=1/2at^2 (solve for a).  Students then change the hanging mass value and measure the new acceleration.  For the hanging mass, I recommend starting around 25 grams and moving up in 25 gram increments.

I ask students which variable is the independent (hanging mass) and which is the dependent (acceleration of cart).  I also ask what variables they think should be held constant throughout the experiment (mass of the cart and distance for which the cart accelerates). I remind students that good practice is to have multiple trials (at least three) per data point. Also, it is essential that students measure the acceleration while the mass is hanging. Once the mass hits the floor, there is no more net force and the cart is no longer accelerating.

I show the 5th slide which has a list of the type of forces we are studying and call on a student to identify the force causing the cart to accelerate. Usually they identify that it is the force of tension from the string that accelerates the cart. The string leads to a mass which is hanging off the pulley. A pulley does not change the amount of force, just the direction of the force being applied. So I ask another student to state which force provides tension on the string. He or she usually correctly identifies gravity pulling on the hanging mass. Slide 7 then shows how to calculate the weight.

## Students Perform Experiment

40 minutes

For this activity, I allow students to group themselves with 3 or 4 in a group. This tends to make the groups homogeneous but for this experiment, that is fine. Students are able to work through this activity successfully no matter the group structure and I give extra support to groups who might struggle. I would love to have smaller groups, but my supplies require this size grouping.  I inform the students that all students will create their own lab write up.

The supplies for the experiment are displayed on the projector and are on the 1st slide of the F=ma experiment Power Point.  For classes that need more guidance and support, I provide a F=ma Lab Write up which outlines the experiment and includes data tables.  Some of my classes, like an honors or AP class, do not require this support and it is enough to show sample data tables (slide 2) to remind them what data to record.

Along with the cart, I instruct students to put mass on cart so that its total mass is 1.5 kg.  The reason for this is that there is friction both in the pulley and on the wheels of the cart.  If the weight of the hanging mass is too small, then the frictional forces will be significant compared to the force from the hanging masses.  If the hanging masses are too large, then the large acceleration of the cart will make it difficult to get an accurate time.  In general, I find that times of less than 0.5 a seconds should be avoided, as the error in the measurement is significant.

As students do the lab, I walk around the room.  Common mistakes I see during the experiment are

• Measuring time and distance after mass hits the floor.
• Large hanging mass accelerates the cart so fast that students cannot accurately measure the time.
• Small hanging mass (less than 20 g) makes the frictional forces significant compared to the force of gravity acting on the hanging mass.
• Incorrect distance measurement as students start the back of the cart at the zero point but stop the time when the front of the cart reaches the end point.
• Students using the mass of the cart instead of the hanging mass.

When I see students making these or any other errors, I ask them a series of questions that lead them to the realization that they need to change what they are doing.  For example, if they let the mass hit the floor while still measuring time, I ask them if the cart is still accelerating after the mass hits the floor.  Once they realize that it is not, they realize they have to change the distance over which they are measuring the time.

## Closure

5 minutes

As the time for the experiment ends, I continue to check students work and make sure they are using the correct masses in their calculation for the net force. When 5 minutes remain, I instruct students to put the supplies away and I display slide 3 of the F=ma experiment Power Point as a reminder for what their write ups should have.  I also have a lab write up template with these parts hanging up in my classroom.

Students complete their write up, graph and analysis for homework. If they need it, I supply some Graph Paper for them to use. I display the last slide which shows what students need to include on their graph. Specifically, I tell students to graph the dependent variable (acceleration of cart) on the x-axis and the independent (applied force/hanging mass) on the y-axis. If students have good data, then they get a linear relationship which can be mathematically analyzed with y=mx+b where y is the force, x is the mass, and the slope of the line is the mass of the cart. The slope intercept, b is zero.  So an analysis of their data shows that F=ma!