Newton’s Second Law

Objectives

Newton’s Second Law, \(\vec{F} = m \vec{a}\), is a cornerstone of classical mechanics. It relates the interactions of a system with its environment — forces — to its motion — kinematics. In this lab, we will measure the motion of accelerating systems and compare these measurements with the predictions of Newton’s Second Law.

For this lab, you should choose two of the following experiments.

  1. Investigate force and acceleration for a cart on a ramp.
  2. Investigate force and acceleration for a cart and pulley system.
  3. Investigate force and acceleration for a pulley system.

You will have your choice among several methods and investigations today. Keep detailed notes. Record your setup, procedure, data, and observations in your lab notes. Data will include tables and plots. Be sure to include block diagrams and sketches of your setup, too. In your memo, summarize your findings and describe how you tackled the design and measurement problems of this lab.

Measuring Force and Acceleration

You will need to measure forces in this lab. The simplest approach is to use the weight of a known mass. If you hang a 100 \(\unit{g}\) mass from a pulley, the applied force will be equal to the weight: \(F = mg = \Quantity{0.980}{N}\). However, pulley systems also involve tension forces. If the system is accelerating, then the tension is not equal to the weight. Keep this in mind when you are comparing your measurements to your theoretical predictions.

There are multiple options for measuring the acceleration. One is to use a ruler and a stopwatch. If an object starts from rest and travels distance \(D\) in time \(T\) under the action of a constant force, then the average acceleration will be \(a = 2D/T^2\). Another option is the motion detectors and Vernier software. A third option is to use the Vernier Video Analysis software. (The most reliable method for determining the acceleration with the motion detector or video analysis is to fit a line to your velocity data.)

Experimental Systems

You can investigate three different systems today. Choose two. If you do a third, you will earn an extra point, but this is not required.

After you take your data — but before you leave, work out the predictions of Newton’s Second Law for the systems you studied. In your memo, compare your data with these theoretical predictions.

As you design your experiments and make your measurements, keep the following two points in mind.

  • Small accelerations are easier to measure than large accelerations.
  • Distances are easier to measure than angles. Use trigonometry to find angles.

Cart on a Ramp

A cart on a ramp will roll downhill due to gravity, but its acceleration is not \(\Variable{g}{9.80}{m/s^2}\). With this system, we can vary the mass of the cart and the angle of the ramp.

Before you take any data, consider the following questions. Try to make quantitative predictions.

  • What will happen to the acceleration of the cart as you increase its mass?
  • What will happen to the acceleration of the cart as you increase the angle of the ramp?

Now collect data.

  • Keep the angle of the ramp constant. Measure the acceleration of the cart for at least three different masses. Record your data in a table and plot acceleration vs. mass.
  • Keep the mass of the cart constant. Measure the acceleration of the cart for at least three different angles. Record your data in a table and plot acceleration vs. angle.

Summarize any trends or patterns you observe in your memo. Compare your measurements with the predictions of Newton’s Second Law.

Cart and Pulley

A mass hanging over a pulley will accelerate a cart on a horizontal track. With this system, we can vary the mass of the cart and the hanging mass.

Before you take any data, consider the following questions. Try to make quantitative predictions.

  • What will happen to the acceleration of the cart as you increase its mass?
  • What will happen to the acceleration of the cart as you increase the hanging mass?

Now collect data.

  • Keep the hanging mass constant. Measure the acceleration of the cart for at least three different cart masses. Record your data in a table and plot acceleration vs. cart mass.
  • Keep the cart mass constant. Measure the acceleration of the cart for at least three different hanging masses. Record your data in a table and plot acceleration vs. hanging mass.

Summarize any trends or patterns you observe in your memo. Compare your measurements with the predictions of Newton’s Second Law.

Atwood’s Machine

Atwood’s “machine” is simply two masses hanging over a pulley. With this system, we can vary each of the two masses, \(m_1\) and \(m_2\) independently. However, in this portion of the lab we will vary the difference and the sum of the two masses separately: \(|m_1 - m_2|\) and \((m_1 + m_2)\).

Before you take any data, consider the following questions. Try to make quantitative predictions.

  • If you keep the total mass constant, what will happen to the acceleration of the system as you increase the difference of the masses?
  • If you keep the difference of the masses constant, what will happen to the acceleration of the system as you increase the total mass?

Now collect data.

  • Keep the total mass constant. Measure the acceleration of the system for at least three separate mass differences. Record your data in a table and plot acceleration vs. mass difference.
  • Keep the mass difference constant. Measure the acceleration of the system for at least three separate total masses. Record your data in a table and plot acceleration vs. total mass.

Summarize any trends or patterns you observe in your memo. Compare your measurements with the predictions of Newton’s Second Law.

Further Exploration

The following activities are not required. If you have the time and the inclination, you may carry out these investigations for extra credit. You are also free to use any of the materials available in the lab to carry out your own investigations. You need to keep a record of your work (setup, procedure, data, analysis) to get credit for your efforts.

  • Additional Experiment: Carry out all three experiments today instead of just two.
  • Alternative Measurements: Repeat one or more of your investigations using a different method for measuring the acceleration. If you used a motion detector, make the same set of measurements with video analysis or a stopwatch. Which method seems more precise?
  • Design your own system. Investigate a combination of ramps and pulleys different than the three described above — for instance, a cart on an inclined plane and a mass on a pulley, or two carts on two different inclined planes connected by a pulley. Be creative, but try to compare your measurements to the predictions of Newton’s Second Law.