How can I use two graphs to approximate g in a lab report?

[haxtor21]

Homework Statement

I am doing a lab report in which I am supposed to plot 2 graphs from which to approximate g. One is Delta X vs t and one is delta X vs t^2. The data represents the time it takes for an object to travel a certain distance interval (in my case a cart going down an inclined plane, and the time being measured by 2 photo-gates) I have SciDAVIs opened and have the data inside 2 tables, but I have no clue how to graph these two tables so that I get the necessary equations for approximating g. All i have right now is scattered data points.

Homework Equations

a=.5gsin(theta) -> g=(2a)/sin(theta) (for graph x-t)

m=.5gsin(theta); g=(2m)/sin(theta) (for graph x-t^2)

The Attempt at a Solution

Im not sure how these equations and graphs will help me get the experimental value of g. Does the computer just spit out some models and I plug them into each other? The lab just says to find the local value of g from both graphs, extremely helpful -_-. Someone please explain me how this is supposed to work. Thank you.

 

[gneill]

What are the expressions for the theoretical curves that you expect? In other words, given an inclined plane at angle Θ to the horizontal, what is the function expressing x versus t?

 

[haxtor21]

What are the expressions for the theoretical curves that you expect? In other words, given an inclined plane at angle Θ to the horizontal, what is the function expressing x versus t?

that would be y= a+ ax + ax^2 and y= bx+c

I figured out the fit for my data points, with the x-t being a polynomial and x-tt being a linear fit. Now what am I supposed to do with them? How do these and my initial equations relate?

 

[gneill]

A cart with frictionless wheel bearings is released from rest on an inclined plane that makes an angle Θ to the horizontal. Derive an expression for the distance x that the cart travels down slope with respect to time t.

Your expression should contain the constant g, the acceleration due to gravity.

 

[haxtor21]

A cart with frictionless wheel bearings is released from rest on an inclined plane that makes an angle Θ to the horizontal. Derive an expression for the distance x that the cart travels down slope with respect to time t.

Your expression should contain the constant g, the acceleration due to gravity.

general form:
x= x-node + v-node (t) + .5(a-x)t^2

in which case a-x is -g

 

[gneill]

What is "node"? How can I calculate x(t) from "node"?

What is the specific equation for x versus t in this particular case? Have you not studied blocks sliding down frictionless slopes? There is a function x(t) = ? which gives the distance that the cart has covered in time t.

 

[haxtor21]

What is "node"? How can I calculate x(t) from "node"?

What is the specific equation for x versus t in this particular case? Have you not studied blocks sliding down frictionless slopes? There is a function x(t) = ? which gives the distance that the cart has covered in time t.

well i guess that would be Δx=.5 (g sin(θ)) t^2

 

[gneill]

Okay! So you would expect your x versus t graph of your data to follow the form

[tex]x(t) = \frac{1}{2}a t^2[/tex]

where a = g sin(θ). If you can find a from your graph, you can find g, right?

What is the slope of the function at some time t = t1? (hint: take the derivative). So pick suitable points along your plotted data and determine the local slope. Use the slope information at point t1 to find g using what you've derived.

For your second graph, where you're plotting x versus t², essentially what you are doing is replacing t² with a new variable τ. That is, τ = t².

Your function becomes:

[tex]x(\tau) = \frac{1}{2}a \tau[/tex]

What's the slope of that graph? It should be easy to determine a and hence g from that one!