Cart on Track With Pulley and Mass Force Sensor

Newton's 2nd Law

Extra Reading:

• Intro to Forces
• Free Body Diagrams
• Momentum Principle (basically Newton's second law)

Newton's second law says:
Where F_net is the total force an object, m is the mass and a the acceleration.

Procedure
The way to test this theory is to apply a force to an object and measure its acceleration. You already know how to measure the acceleration - using the sonic ranger and making a graph. But, how do you measure the force applied to the object and how do you keep this constant? It is possible to measure a force with a force probe, but how do you keep it constant so that you can measure acceleration? To do this, you will use a set up such as the one below.

Analyzing this motion may seem difficult, but it is not. Think of this situation by answering these questions:

• What is the total mass being accelerated? Well, both M₁ and M₂ are being accelerated, so the mass being accelerated is M = M₁ + M₂.
• What is the force accelerating the system? There are several force acting on this system, but the only one that is not cancelled and leads to an acceleration is the weight of M₂, so F = g*M₂

So, to summarize, we have a mass M = M₁ + M₂ being accelerated by the weight of M2. Thus you can measure the acceleration and the force is known.
In scalar form:

or in this case

In this experiment, you want to change the force and measure the corresponding acceleration. To do this, M₂ will be a "mass hanger" with mass on it. If you put less mass on M₂ it will reduce the force - but this would also change the total mass (which we want to keep constant). The solution is to take mass off of M₂ and add it to M₁, so that M₁ + M₂ is always constant.
Technical Details:

• Start with 0.12 kg for M₂. This will consist of the hanger (which is .05 kg) and several 10 gram masses.
• For the cart, add two of the 500 gram bars. You will need the total weight of your system - you can do this by putting the whole thing (cart, masses, hanger) on the balance - but this may be over the max limit for the scale. If this is the case, you can determine the mass of the individual pieces, then just add to get the total mass.
• To find the acceleration, put the sonic ranger at the end of the track. You are then going to take data while the cart is moving towards and away from the sonic ranger.
• The trick is to start recording data, and then gently push the cart towards the sonic ranger. Make sure it does not get too close, or it will not give good data. Stop the cart with your hand before it crashes into the pulley. Also make sure all your "stuff" is lined up - meaning the string and the pulley and the cart.
• To find the acceleration - just fit an equation to the selected part of the graph. Remember it should be a 2nd order polynomial and the acceleration will be twice the number in front of the t² term.
• MAKE SURE your track is level. If it is not level, it is not such a simple problem. (I know the tables are NOT level).
• Make sure that the string connecting the cart is parallel to the track. If it is not, you will only have a component of the tension making the cart accelerate.
• Make sure the pulley is not tilted, if it is there will be significant friction.
• hint: To select the appropriate section of data, it may be easier to look at the velocity graph instead of the position graph.

Uncertainty in the acceleration measurement

• Hopefully, you noticed that doing error propagation for the acceleration gives you a very large uncertainty for the acceleration. The problem is that this assumes you are only looking at three data points at a time. When you add a trendline to get the acceleration, you are looking at all data points at once.
• For this lab, you will need to determine an uncertainty for the acceleration. To do this, you will measure the acceleration several times (at least 5). The uncertainty will then be found from the standard error. Detailed instructions on finding the standard error are located in the uncertainty page.
• For the uncertainty in the mass - you will have to make an educated determination (guess).

Things to think about

• Why is it a good idea to measure the acceleration of the cart for a trip that consists of both directions (i.e. why not just let it go and measure the acceleration as it moves towards the pulley?
• If Newton's second law works, what SHOULD your graph of Force vs. acceleration look like?
• If Newton's second law works, what should the slope of your graph be?
• If you data does not fit - what are some possible explanations?

Part II
What would be different if you raised the track so it was not level? See if you can calculate what the acceleration should be for a given mass and angle. Then measure the acceleration to see if you get the same value.

Homework
1. Suppose a cart of mass 0.5 kg is placed on a track and connected to a mass on a pulley hanging over the edge. If the hanging mass is 0.2 kg, what would the acceleration of the cart be?
2. How are you going to measure the acceleration of the cart?
3. How are you going to determine the uncertainty in the acceleration of the cart?
4. In order to explore the relationship between force and acceleration, what do you have to keep constant in this lab?
5. The lab instructions suggests to measure the acceleration of the cart as it goes one direction and then changes directions. Why do you think this is a good idea?

Cart on Track With Pulley and Mass Force Sensor

Source: https://www2.southeastern.edu/Academics/Faculty/rallain/plab193/page1/page27/page27.html

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