ATWOOD MACHINE and INCLINED PLANE

PART 1) ATWOOD MACHINE

 The Atwood machine is a low friction fixed pulley with two masses m1 and m2 hanging at the ends of a thin string that passes around the pulley. Of course, if m1=m2 the system is in equilibrium and should remain at rest if initially at rest. Ideally, a small difference in masses (for instance by transferring a small mass from one side to the other) will produce an unbalance and the system will accelerate according to the following expression:

In practice, there is a minimum mass difference that will start the motion (due to static friction) and there is always some dynamic friction force f (even with our "low friction pulley").

So a more realistic model for the Atwood Machine's dynamics would be :

F_net = m a => (m₁ - m₂) g - f = (m₁ + m₂) a

so that:

(m₁ - m₂) g = (m₁ + m₂) a + f

You will measure (using a motion detector placed on the floor) the acceleration a for 5 different values of the mass difference keeping the total mass (m₁ + m₂) constant.

A plot of the "applied force" (m₁ - m₂) g as a function of the measured acceleration a should be approximately linear with slope (m₁ + m₂) and intercept f. Call this value for the frictional force f1. Using LINEST find the slope and intercept with their uncertainties.

Compare the slope with the actual total mass (m₁ + m₂).

Do they agree?

To obtain an independent determination of the frictional force you will place the system in equilibrium (50 g on each side) and give a small push. From the position data you should be able to obtain the (small) deceleration due to friction and from that the frictional force. Call this value for the frictional force f2.

Compare f1and f2.

Do they agree?

Discuss if air resistance may be neglected in this experiment or not. Use physical reasoning.

PART 2) INCLINED PLANE

Use the Pasco metal tracks and low friction Pasco carts to study the dependence of the acceleration a of the cart with the angle of inclination α of the track (use five angles: 3, 6, 9, 12 and 15 degrees)..

You will determine the inclination with the angle finder and/or measuring the lengths and using appropriate trigonometric relations, and the acceleration using the motion detector. Use the electronic scale to measure the mass of the cart (include the wood block!).

Also you will determine the force Fappl that you need to apply parallel to the track using a Force probe to keep the cart from moving down the track. A plot of Fappl as a function of sin α should be linear with slope mg and intercept 0.

NOTE: If you use Excel to calculate sin α, you need to convert the angles from degrees to radians. For instance, to obtain sin (3°) you would type: =SIN(RADIANS(3))

NOTE: To obtain the acceleration values for both Part 1 and Part 2 you may use in LoggerPro a linear fit of the v-t data (slope=a) or a quadratic fit of the x-t data (coefficient of the quadratic term = 1/2 acceleration ).

For an inclined plane with friction a = g(sin c - mu cos c) where c is the base angle and mu is the coefficient of friction.

For c ~15 degrees or less cos c ~ 1 then a = g sin c - mu g

A plot of a as a function of sin c should be linear with slope g and intercept mu g. Use LINEST to get the uncertainties.

Does the slope agree with the expected value?

Is the coefficient of friction significantly different from 0?

Do the slope and intercept agree with the expected values?