Cart and hanging mass lab report answers

Lab 3: Newton’s Second Law of Motion

Introduction

Newton’s Second law of motion can be summarized by the following equation:

Σ F = m a (1)

where Σ F represents a net force acting on an object, m is the mass of the object moving

under the influence of Σ F, and a is the acceleration of that object. The bold letters in

the equation represent vector quantities.

In this lab you will try to validate this law by applying Eq. 1 to the almost frictionless

motion of a car moving along a horizontal aluminum track when a constant force T

(tension in the string) acts upon it. This motion (to be exact the velocity of the moving

object) will be recorded automatically by a motion sensor. The experimental set up

for a car moving away from the motion sensor is depicted below.

If we consider the frictionless motion of the cart in the positive x-direction chosen in

the diagram, then Newton’s Second Law can be written for each of the objects as

follows:

T Ma (2)

and

– gT F ma  (3)

From this system of equations we can get the acceleration of the system:

2

gF

a m M

 

(4)

Because the motion of the car is not frictionless, to get better results it is necessary to

include the force of kinetic friction fk experienced by the moving car in the analysis.

When the cart is moving away from the motion detector (positive x-direction in the

diagram) Newton’s Second Law is written as follows for each of the moving objects

m and M:

1 1– kT f Ma (5)

and

1 1– gT F ma  (6)

Since it is quite difficult to assess quantitatively the magnitude of kinetic friction

involved in our experiment we will solve the problem by putting the object in two

different situations in which the friction acts in opposite directions respectively while

the tension in the string remains the same.

When the cart M is forced to move towards the motion detector (negative x-direction

in the diagram), the corresponding Newton’s Second Law equations will change as

follows:

2 2kT f Ma   (7)

and

2 2gT F ma  (8)

Note that in equations 5, 6, 7, and 8 the direction of acceleration represented by vector

a has been chosen in the same direction as the direction of motion.

We are able to eliminate the force of kinetic friction on the final result, by calculating

the mean acceleration from these two runs:

 1 2

2 ave

slope slope a

  (9)

Combing the equations (5) – (8) we derive the equation to calculate the value of

gravitational acceleration:

 avea M mg

m

  (10)

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Equipment

Horizontal dynamics track with smart pulley and safety stopper on one end; collision

cart with reflector connected to a variable mass hanging over the pulley; motion

detector connected to the Science Workshop interface recording the velocity of the

moving cart.

Procedure:

a) Weigh the cart (M) and the small mass (m) hanger. b) Open the experiment file “Newton 2 Law”: Desktop: pirtlabs/ PHY 122 PreSetUp

Labs/Newton 2 Law.

c) Pressing the START button you will initiate data recording. It stops automatically after 5 seconds. Make a trial run to check if any adjustments to the motion sensor

are needed.

Experiment 1

The objective of this activity is to verify the relationship between net force and

acceleration for each of the accelerating objects in the system as described by equations

(2) and (3). For this we will use equation (4), which is the result of the combination of

these two equations. We will keep the mass of the system constant (m+M) and vary

the gravitational force on the hanging mass. According to equation (4), the acceleration

of the system should change linearly with this force.

1. For Run # 1, arrange four 10.0 g masses on the cart so that they are evenly distributed; hang another 10.0 g mass on the S-shaped hanger. Pull the cart along

the track as far as possible away from the pulley but not closer than 15 cm to the

motion sensor. Press Start and release the cart. Stop the car by hand before it bangs

into the magnetic stopper mount at the end of the track. Apply linear fit to the part

of the data that is not in immediate proximity of the magnetic stopper. Record the

slope. What physics quantity does the slope of this velocity vs time graph

represent?

2. Repeat the above steps to make four additional runs. For each consecutive run, remove a 10.0 g mass from the cart then place it on the hanger. No matter how

many 10.0 g masses remain on the cart always try to keep the masses on the cart

symmetrically distributed. Record slopes of the graphs for each run. Remember the

physics meaning of the slope of velocity vs time graph.

3. In Logger Pro plot acceleration of the system as a function of the force of gravity on the hanging mass. Apply linear fit to the acceleration vs force of gravity graph.

Find the experimental value of the mass of the system along with its uncertainty by

using the parameters of the regression line of acceleration vs force of gravity

graph. The experimental value will be compared to the direct measurements of the

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mass of the system from the scale. Does your plot show any evidence of friction?

How can you tell?

Experiment 2

In this experiment you will determine the acceleration due to gravity by negating the

effect of friction for opposite directions of motions. The correct value of gravitational

acceleration should confirm the validity of application of the Newton’s Second Law.

You will also use a statistical approach to the experimental data.

4. In Logger Pro set a new data table with the following new columns: “m” (kg), “slope 1” (m/s2), “slope 2” (m/s2). Notice that “m” represents the total hanging

mass responsible for tension in the string (the slotted weights and the S-shaped

hanger). The new calculated columns “aave” (m/s2), “g” (m/s2) should be also

created: (DATA  NEW COLUMN  CALCULATED) according to equations

(9) and (10).

5. For 6 values of tension in the string ranging from 0.1 N to 0.4 N excluding the weight of the hanger, plot velocities vs time graph of the cart moving toward

and away from the motion sensor. To put the cart in motion give it a careful push

towards the motion sensor after starting the recording.

6. In Logger Pro record the value of the slope of the cart when it was moving toward to the motion sensor (slope 1) and the value of the slope when the cart was

moving away from the motion sensor (slope 2).

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7. Insert a new graph “g vs g” use the statistics button from the top menu bar to get a mean value of the experimental g and its standard deviation (uncertainty).

Always manually bring the cart to a full stop before it hits the magnetic safety

stopper!

The data with negative velocities represents the motion towards the sensor. In this case,

kinetic friction acts in the same direction as the tension generated by the hanging mass

hence the absolute value of acceleration is relatively bigger (slope 1 = a1) compare to

the value of acceleration when the cart moves away from the motion detector.

When selecting the data for linear fits to each direction of motion avoid the points

corresponding to the end of the track where the magnetic stopper is mounted.

For the results of this lab you need to state in experiment 1 the experimental value of

the mass of the moving system, its error and theoretical value of mass determined with

the balance.

You also need to state the mean value of “g” calculated in Logger Pro along with its

uncertainty (standard deviation). How does this experimental result compare to the

expected value of 9.81 m/s2? Calculate the discrepancy between expected value of g

and your experimental results:

exp % 100%

theor

theor

g g discrep

g

  

Be sure to report the results with the correct number of significant figures.

In the Discussion and Conclusion section answer the questions included in the lab

procedure. Explain the relationship you recorded between acceleration and force. Also

explain why the slope has slightly bigger value when the cart moves toward motion

sensor compare to when it moves away. Explain the nature of errors and reasons for

discrepancies between your experimental and theoretical results. Based on your

experimental data from both parts state whether Newton’s Second Law of Motion was

verified.