Physics 11 Gizmo – Atwood Machine
The Atwood Machine was invented by the English physicist George Atwood in 1783. Atwood was a popular lecturer at Cambridge who loved to demonstrate physical principles using pendulums, pulleys, pumps, and many other devices. A natural outgrowth of this passion was his machine, which is used to measure acceleration and demonstrate Newton's Laws. The Atwood Machine, which is simple enough to build in any lab, has remained a classic physics demonstration.
Qualitative Relationships
In this activity, you will investigate the relationships between the masses of two hanging weights and their resulting motion. Remember that in controlled experiments, only one variable is changed at a time.
1. Observe the Atwood Machine on the SIMULATION pane of the Gizmotm. In the default setting, masses of 2.0 kg and 3.0 kg hang from a 2.0 kg pulley with radius 0.2 m. On the DESCRIPTION pane, the force vectors are shown for each mass. The longer the arrow, the greater the force it represents.
a. Based on the red arrows, which is greater, the force pulling Mass A up or the force pulling it down?
b. Based on the blue arrows, what is the direction of the greatest force on Mass B, up or down?
c. Click Play () and observe the SIMULATION pane. In which direction did each mass move?
d. Note the time displayed on the CONTROLS pane at lower right. How long did it take for Mass B to reach the bottom? Record this time in your notes.
e. Select the TABLE tab and look at the velocity numbers. Notice that the velocity of Mass B is negative, which means that Mass B is dropping down. As time goes by, what happens to the velocity of each mass? Objects that are increasing or decreasing in velocity are said to be accelerating.
f. Compare the velocity numbers for Mass A and Mass B. At any given time, what is the relationship between the velocity of Mass A and the velocity of Mass B?
g. Select the BAR CHART tab, and check Show numerical values. Click Reset (), and then Play again. What did you notice about the velocity of Mass A and Mass B? What was the final velocity attained by Mass A? Record this value in your notes.
2. Click Reset. You can use the sliders on the SIMULATION pane to change the mass of Mass A or Mass B, as well as the Radius and Mass of the Pulley. Perform several experiments and record the following: starting conditions, time to complete the descent, and the final velocity of each object. As you experiment, try to answer the following questions:
a. What is the effect of increasing the difference in mass between Mass A and Mass B? What happens to the time and final velocity when Mass A and Mass B are nearly identical?
b. What is the effect of increasing the mass of the pulley?
c. What is the effect of changing the radius of the pulley?
3. Set the Pulley mass to 0.0 kg. Record the final velocity and the time for Mass B to drop for the following two situations:
a. Mass A = 2.0 kg, Mass B = 3.0 kg
b. Mass A = 4.0 kg, Mass B = 5.0 kg
c. In each experiment, the difference in masses was 1.0 kg. Were the final velocities and completion times the same? Explain.
d. Why do you think this is?
Calculating Force and Acceleration
The Atwood Machine was invented to demonstrate Newton's Laws. Newton's Second Law states that force is equal to mass times acceleration:
F = ma
Each of the masses experience two forces. Gravity pulls each object down, and the tension on the string pulls each object up. In this activity, you will calculate these forces and determine the acceleration of each mass.
1. Click Reset, and set Mass A to 2.0 kg and Mass B to 3.0 kg. To simplify calculations, select Frictionless on the SIMULATION pane. Click Play and wait for the simulation to finish.
a. Choose the GRAPH tab and observe the velocity graph at bottom. If the acceleration is constant, the velocity curves will be linear (straight lines). Is the acceleration of each object constant in this case?
b. Select the BAR CHART tab. What is the final velocity of Mass A?
c. When an object starts at rest, the average acceleration is equal to the final velocity divided by time. Use a calculator to determine the average acceleration of Mass A, and write this number down (you will need it later). The units of acceleration are meters per second per second, or m/s2.
2. Select the DESCRIPTION tab. For each individual object, the downward force is equal to the object's weight, which is the product of mass and gravitational acceleration. (w = mg, where g = 9.81 m/s2 on Earth.) The unit of force is the Newton (N), equal to the force needed to accelerate a 1 kg object at a rate of 1 m/s2. For example, the weight of Mass A (which has a mass of 2 kg) is 2 • 9.81 = 19.62 N. In this case, Mass A would exerta force equal to 19.62 N on the rope.
a. What is the weight of Mass B? To check your answer, click Show numerical values. The weights of each object are displayed below the objects.
b. Because Mass A and Mass B are connected by a rope, the two objects are considered to be linked. What is the total mass of the two objects?
c. Notice that the force of gravity on Mass A counteracts the force of gravity on Mass B. When two forces on an object are opposed to each other, the total force is the difference of the two forces. For example, if Mass A had a weight (or force) of 6 N and Mass B had a weight of 10 N, then the total force on the system would be 10 - 6 = 4 N. What is the total force on the Mass A Mass B system shown in the Gizmo?
d. Newton's Second Law states that F = ma. Therefore, a = F/m. To calculate the acceleration for this system, divide the total force you calculated in step c by the total mass you calculated in step b. Is this the same as the acceleration you determined in part 1c above? (Note: The numbers may not be exactly the same due to rounding errors.)
3. The next task is to derive an equation for the acceleration of any set of masses, mA and mB. For simplicity, assume that mB is greater than mA. Recall that the force in the direction of motion is the weight of object B, and the opposing force is the weight of object A. The equation for weight is w = mg, where g = 9.81 m/s2 on Earth.
a. What is the total force on the system? Write an expression using the variables g, mA and mB (Hint: subtract the smaller weight from the larger weight.)
b. What is the total mass of the system?
c. What is the acceleration? (Use F = ma and solve for a.)
d. Use the Gizmo to test your equation using several values for Mass A and Mass B. Recall that acceleration is the final velocity divided by the time. If your equation gives incorrect results, recheck your math and consult your classmates and teacher, if possible.
4. On the SIMULATION pane, choose the Frictional option and set the Pulley mass to 5.0 kg. Use the default settings Mass A = 2.0 kg and Mass B = 3.0 kg. With friction, the rope causes the pulley to turn rather than just sliding along it. Because additional energy is required to turn the pulley, the acceleration of the two masses is slowed.
a. Using Newton's Second Law, we have F = ma, or a = F/m. Does the pulley mass add any additional force to the equation? In other words, does the weight of Mass A or Mass B change?
b. If the acceleration is changed but the force is the same, then there must be a change in the mass that is being accelerated. When the pulley is factored in, it adds the equivalent of half of its mass to the system. (This assumes that the pulley is a solid disk of uniform density.) The resulting equation is
Where g is the gravitational acceleration on Earth (9.81 m/s2), mA is Mass A, mB is Mass B, and mP is the mass of the pulley.
c. Experiment with several values for the pulley mass. Does the experimental acceleration of the objects match the acceleration predicted by this equation?