Atwood machine
Study Newton's second law (F = ma) by measuring the acceleration of a cart connected to a suspended mass through a pulley.
The Atwood machine, invented by mathematician George Atwood in 1784, is one of the most elegant devices for experimentally verifying Newton's second law. By connecting two masses through a pulley, the system accelerates at a rate that depends on the mass difference. With a smartphone on the cart and its accelerometer recording the motion, you can measure the acceleration directly and verify the theoretical prediction a = m₁g/(m₁ + m₂).
Learning objectives:
The student places their smartphone on a cart connected by a string to a suspended mass via a pulley. By varying the suspended mass, the student verifies Newton's second law.
Level:
High School
FizziQ
Author:
Duration (minutes) :
30
What students will do :
- Measure the acceleration of a mechanical system with the smartphone accelerometer
- Experimentally verify Newton's second law (F = ma)
- Plot and interpret the acceleration versus hanging mass graph
- Identify the sources of discrepancy between theory and experiment
- Understand the role of total system mass in determining acceleration
Scientific concepts:
- Newton's second law (F = ma)
- Mass and acceleration
- String tension
- Ideal pulley
- Constant acceleration
- Net force
Sensors:
- Accelerometer (linear acceleration)
What is required:
- Smartphone or tablet with FizziQ
- A low-friction cart (or board with wheels)
- A pulley fixed at the edge of a table
- String and hanging masses (20-200 g)
- Adhesive tape to secure the phone
Experimental procedure:
Set up a horizontal track (air track, or simply a smooth table) with a pulley or guide fixed at the edge.
Place the smartphone on the cart (or directly on the table if smooth enough) and secure it with adhesive tape.
Attach a string to the cart. Run the string over the pulley and hang a mass m₁ from the free end.
Open FizziQ and select the Accelerometer instrument (linear acceleration in the direction of motion).
Hold the cart still, start the recording, then release the system.
Record the acceleration during the cart's displacement (a few seconds is enough).
Note the constant acceleration measured (ignoring the initial release spike and the final impact).
Calculate the theoretical acceleration: a = m₁g / (m₁ + m₂), where m₂ is the mass of the cart + smartphone.
Repeat the experiment by modifying the suspended mass m₁ (in increments of 20 to 50 g). Plot a versus m₁.
Verify that the graph a = f(m₁) is a curve that approaches g when m₁ becomes much larger than m₂.
Expected results:
The measured acceleration is constant during the motion (after the initial release transient). The measured value is slightly less than the theoretical prediction due to friction. The graph a(m₁) is a curve that increases from near zero and asymptotically approaches g. Typical measured values range from 0.5 to 5 m/s² for reasonable mass ratios.
Scientific questions:
- Why is the measured acceleration always slightly less than the theoretical value?
- What happens when the suspended mass becomes very large compared to the cart mass?
- How does the pulley's mass affect the acceleration?
- Why is the string tension during motion less than the weight of the hanging mass?
- How could you use this experiment to measure g?
- What assumptions of the ideal model are not met in practice?
Scientific explanations:
Newton's second law states that the acceleration of a system is proportional to the net force and inversely proportional to the total mass: a = F_net / m_total.
In the modified Atwood machine, the only driving force is the weight of the suspended mass: F = m₁g. The total mass in motion is (m₁ + m₂), giving a = m₁g / (m₁ + m₂).
If m₁ is very small compared to m₂, the acceleration is small and proportional to m₁ (linear regime). If m₁ is very large compared to m₂, the acceleration approaches g (free fall limit).
The graph of 1/a versus (m₁ + m₂)/m₁ is a straight line with slope 1/g. This linearization is a classic technique for verifying inverse relationships.
Sources of discrepancy with theory include: cart friction on the track (which reduces acceleration), pulley mass and friction (which add effective inertia), and string mass.
George Atwood invented this device in 1784 because the chronometers of the time were not precise enough to measure the acceleration of free fall directly. The machine 'dilutes' gravity, making the motion slow enough to time.
The tension in the string is not equal to the weight of the suspended mass during motion. It equals T = m₁m₂g / (m₁ + m₂), always less than m₁g.
This experiment is a textbook case of Newtonian mechanics: the system is simple enough to be solved analytically, yet rich enough to illustrate the fundamental principles.
Extension activities:
- Why is the measured acceleration always slightly less than the theoretical value?
- What happens when the suspended mass becomes very large compared to the cart mass?
- How does the pulley's mass affect the acceleration?
- Why is the string tension during motion less than the weight of the hanging mass?
- How could you use this experiment to measure g?
- What assumptions of the ideal model are not met in practice?
Frequently asked questions:
Q: The acceleration is not constant on my graph.
R: Friction may vary along the track. Use the most regular part of the graph. Also check that the string does not rub on the table edge.
Q: My calculated g from the data is too low.
R: Friction and pulley inertia reduce the measured acceleration. Account for friction by estimating the deceleration when the cart is given a push without the hanging mass.
Q: The phone shifts on the cart during the experiment.
R: Secure it firmly with adhesive tape. The phone must move as one unit with the cart.
Q: Can I use a toy car instead of a lab cart?
R: Yes, but ensure the wheels turn freely. Measure the phone + car mass accurately.