Works acting on a spring system

In this lab, we were going to study about the energy in spring system.
As the picture shows, we set the spring holding a mass. The motion sensor was below the mass for measuring the velocity and the position of the mass. The force sensor hold the spring for measuring the elastic coefficient.

 Therefore, we need to figure out what kinds of energy will be exist on the spring, and how they interact together.

1. Kinetic Energy
2. Static Energy
3. Elastic Potential Energy

First of all, we need to know the kinetic energy of the mass. We apply the equation with mass and velocity:

And then, we need the static energy and apply the equation with the velocity and mass:

At the last one, we also need to know the elastic potential energy and apply the equation:

However, we still did not know the elastic coefficient k. Thus, we measured the position of the non-stretched mass and the force applying to the force censor  And then, we added some mass to the spring so that the spring could be stretched longer, and we measured the new position and the new force applying to the force censor. Therefore, we could use this equation to get k.

And we can use this to find the elastic potential energy. And then, we make a table to record the number and a graph to show how they changing. For readability, we make a parameter for total energy.

From those pictures,we could discover that the total energy was a constant even though it was not perfect, so we can say the even though the energy always change, the total energy is unchangeable except there is another object applies work on it.

Work and Kinetic Energy

The lab we just did today is finding the relationship between work ad kinetic energy.
First of all, we need to set up a modal to find the equation for kinetic energy.

As the pictures show above, we connected a car and the force sensor that is on one side of track with spring and set the motion sensor on the other side, so we were able to measure velocity of car and force that applied on the car. And then, we put a wooden block on the track and put the spring on the block in order to reduce the friction from track to spring. 

For studying the work and kinetic energy, we need to know how much work the car had done.

we created a graph that represent the force and car's position.

Therefore, the area below line is work the car had done.

Now, we also need to create a line that describes kinetic energy:

As what we can see, the work line and kinetic energy line is almost overlapping together. Thus, the relationship is work=kinetic energy.

Work and Energy

In this lab, we were going to study about work and energy.

we went out side to calculate the amount of work for walking up in height h with stair and how much work we use to drag an object up the same height. We will also calculate the power as well.

We had three steps we need to do in lab.

1.drag a bag with mass of 9 kg and record the time we used.

2.walk up to the same height and record the time we used.

3.run up to the same height and record the time we used.

From the equation, we can calculate the work to apply to finish our work.

W=Fd

First,we need to know the height of the floor.

we measure each stair height is 0.17m, and there are 26 steps, so the total height is 4.42m.

And then, we measured the time for dragging object, walking, and running.

dragging: 15.45s

walking: 16.13s

running: 4.77s

dragging:

Work:

(9kg) (9.8m/s^2)(4.42m) = 389.844 joules

Power:

work / 15.45 second = 25.23 watts

walking:

Bag on pulley:

(95kg)(9.8m/s^2)(4.42m) =  4115.02 joules

Power:

work / 16.13 seconds = 255.11 watts

running:

Bag on pulley:

(95kg)(9.8m/s^2)(4.42m) =  4115.02 joules

Power:

work / 4.77 seconds = 862.68 watts

centripetal acceleration as a function of angular speed

In this lab, we were going to study about the angular velocity and angular acceleration.

As the picture show, we would rotate the plate and collect the period with timer. And then, we got following graph.

From the graph, we can see when period decreases, the acceleration is increasing.

Moreover, we know the equation:

w=2Pi/T

Therefore, when angular speed increases, the acceleration is increasing.

And we also found the graph about angular acceleration and angular velocity^2 gives a positive slope of 0.1493 as liner function. 

It shows that both of them are proportional to each other, and because it matters with radius of the turn table because a=r*w^2. a is acceleration and w is angular velocity, r is the slope of the graph.

Frind relationship between angular velocity and angle of a conical pendulum

In the lab, our purpose is finding the relationship between the angle of conical pendulum and angular velocity.And this is the picture showing about lab:

And then, we use the ruler to measure l ans L.
l: 0.6 m L: 0.67 m H(total height) : 2.3 m
However, what should we do for measure the angular velocity? And what can we measure the angle during the object is rotating? We will get not accurate number if we only use angle measure tool and angular velocity measure tool. Thus,we can use equation to instead it to  measure the angle and angular velocity.

Angular velocity = 2*Pi/period

Thus, after measuring data, we got those number:

1. T1=3.968s h1=0.5m
2. T2=3.288s h2=0.628m
3. T3=2.953s h3=0.762m
4. T4=2.698s h4=0.955m
5. T5=2.514s h5=1.066m
6. T6=2.205s h6=1.14m
7. T7=1.99s  h7=1.425m
8. T8=1.85s  h8=1.487m

From those number, we use equations to get the graph:

Now, the goal of this lab is finding the relationship, so we made some equations:

divide both side with m:

Because the r is changing with the angle, so we wrote the equation between r and angle:

we apply those two equations, and we can got:

So we use Excel to get the angular velocity by the angle:

From this graph, we can figure out that theory w and real w are similar, so the equation about angle and angular velocity is correct.

Modeling Friction Forces

In this lab, we study about friction of object and figure out coefficient of static friction. And we built 4 different model about friction of object.

1.Put a wooden black on the table, tie a string to the block and connect to water cup over a pulley at the end of the track.And then, we put water to the cup patiently until the block finally starts to slide to edge of table.Record the mass of the cup and water required to get block to start to move.And we add 3 times wooden block and record them to record their mass of wooden block and mass of water.

 The record :

After this, we plug all data into computer to create the graph:

So that, we can the coefficient of static friction between table and block is 

0.2188.

2.drag the block in constant speed by motion sensor which is connected to the block, and we get the graph from Labpro:

And this graph gives us the average kinetic friction force between the block and the track.

After we measure those blocks' mass, we create the table to record mass of block and normal force.

      mass(kg)      normal force(N)   friction(N)

1.  0.11          1.078              0.3177
2.  0.236         2.3131             0.6187
3.  0.295         2.89               0.852
4.  0.45          4.411              1.3

we model sliding friction as being proportional to the normal force, and speed of the moving object. That is, we create a function:

Fkinetic=K*N=>K=Fkinetic/N

we get the number K is 0.2947. 

3.We make a slope with the metal track and put the motion sensor at the bottom of the track, and then let a wooden block to slide down to bottom from top.And we can record the acceleration with computer and sensor, show in picture:

from this graph, we are able to apply the equation to get net force:

and the net force is equal to sin(a)*mg-Ffriction so we can get:

=>

divide the m

=>

the a=0.8527m/s^2,angle a=20degrees, so we get k=0.2807

4.we make a slope, connect the block which is on slope to a mass over pulley, and make the angle between slope and horizontal table to be just make the mass can drag the block to move.
And now, we know the mass is 0.15kg,angle is 21.5degree,block is 160kg, and acceleration is 0.545m/s^2 with measuring.
we create a equation:

and we can get k is 0.5539

measure density of metal

In this lab, our goal is figuring out the uncertainty of values.
In the first part, we were going to measure 3 different metals' length, diameter, and mass in order to get the density with its uncertainty.

And then, we get:

    metal         length        diameter         mass            density

1. steel      5.00+-0.01 cm  1.26+-0.01 cm  48.9+=0.1 g      7.6 g/cm^3
2. copper     5.14+-0.01 cm  1.28+-0.01 cm  58.4+=0.1 g      8.8 g/cm^3
3. brass      4.80+-0.01 cm  1.60+-0.01 cm   80.0+=0.1 g     8.3 g/cm^3

From the equation for density:

And then, we take the natural logarithm of each side we get:

=>

1. steel 
   
2. copper
   
3. brass
   

This,we can get those metals' density with their uncertainty:

1. steel:  7.6+-0.15 g/cm^3
2. copper: 8.8+-0.16 g/cm^3
3. brass:  8.3+-0.12 g/cm^3

And, there is next lab to determine the mass of object by measuring two angles and those forces of two strings.

All of first, we focus on the y size of the object.

and then, we focus on the x size of the object.

The data we measure about force on strings and angles:

F1=7.5+-0.5N

F2=7.0+-0.5N

a1=46degree+-1degree

a2=50degree+-1degree

plug those data to equation.

And we can get the force from equation

is 1.0976+-0.015N