Please follow the instruction clearly

As a natural next step from Lab 9 (Conservation of Mechanical Energy), let us consider the small but non-zero friction.

https://openstax.org/books/college-physics/pages/7-introduction-to-work-energy-and-energy-resources

We have two ways to determine mu (of frictional coefficient); compare them.

Following the basic report format (from intro to conclusion), submit a report as a single pdf file, including a completed spreadsheet.

The lab format should be something like this

Please follow the lab format (Title, Objective/s, Introduction, Results & Discussion, and Conclusion).

-LAB Format?

Objective/s:

The purpose/objective of this experiment is to?.

-Introduction:

Write 5 to 7 lines to introduce the topic of measurements. You may add any used formula/s in this section. You may insert the experimental setup in this section.

-Results & Discussion:

Insert your data, numbers ? pictures ? table- in this section. I also need sample calculations (as an example; details of how you got the area or volume of a soup can in cm and inch with the right units).

-Conclusion/s:

Please be specific and reflect on the results you got, numbers/data, from the Results & Discussion section into this section (conclusion).

– Conclusion should be a short section roughly 1 to 3 lines will do the job. Just make it clear.Work Energy Relation on Inclined Plane with Friction

Watch the movie at the link below, in which a wooden block slides down an inclined surface.

In the first part, we find the angle at which a small and quick push makes the block slide down at a constant, slow speed. The coefficient of kinetic friction is the tangent of this angle.

In the second part, measurements are taken for the block sliding down at larger angles. A photogate is placed near the top of the incline and another near the bottom. The time to pass through each photogate is measured so that we can find the velocity at both locations. These measurements can also be used to find the coefficient of kinetic friction.

We use the block sliding down the incline, in the presence of friction, to illustrate the relationship between work and mechanical energy.

(A different lab manual is referred to occasionally in the video. Those instructions are not relevant to you. However, all the measurements are the same.)

https://cuny907-my.sharepoint.com/:v:/g/personal/darya_krym87_login_cuny_edu/Ebb_izhDDqJFpbWCJ7VUF3gBDABjARSu4MWbeygi_Vv7bw?e=5pTClw

Description of Experiment

In the first part of the experiment, we try to find the angle such that the parallel component of the gravity force, is equal to the friction force, so that the total acceleration is zero and the block descends at a constant speed, once it is in motion. However, we cannot simply look for the angle at which the block starts to slide, because before moving, the block experiences the force of static friction, which is larger than the kinetic friction force.

A small and quick push is needed to overcome the initial static friction force. This gives the block a small initial velocity and puts it in the kinetic friction regime. The push should be quick and small because we want the block to have a slow initial speed, since in this case, it is more obvious if the block?s speed is not constant.

In the second part of the experiment, the angle is large enough that the block accelerates along the inclined plane. The parallel component of the gravity force is larger than the kinetic friction force. It is even larger than the static friction force so that the block starts sliding from rest.

Theoretical Principles

In this lesson we are going to generalize a couple of topics we covered previously, friction and conservation of mechanical energy. As in the previous lesson, we consider an object moving along an incline, so that the gravity force is partially along the direction of motion and partially perpendicular to it. However, this time we also examine the effect of friction.

Friction

In lesson 6, we examined the friction force, . For an object in motion, we discuss the kinetic friction force, whose magnitude is proportional to the magnitude of the normal force, , and the coefficient of kinetic friction, , which depends on the materials of the surfaces in contact

(1)

(Often, a subscript is used to specify the coefficient of friction is the kinetic one. We omit the subscript here since we only talk about kinetic friction i.e. friction during motion, in this lesson.)

Recall that the normal force, , is the force of a surface acting on an object, in the direction perpendicular to the surface (e.g. the force of the floor or of the chair, on you, as you stand or sit). The direction of the kinetic friction force is opposite the direction of motion.

To find the normal force, we must consider the forces and acceleration (if present) in the direction perpendicular to the surface. For an object on a horizontal surface, the forces perpendicular to the surface are gravity and . There is no acceleration perpendicular to the surface so these two forces balance each other. To find for an inclined surface, we, still, consider the forces perpendicular to the surface. Now, we have and , where ? is the angle between the surface and the horizontal. No acceleration in the direction perpendicular to the surface[ Lack of acceleration perpendicular to the surface is simply the statement that the object is staying on the surface, as opposed to crashing through it or flying away from it. For an object to accelerate perpendicular to the surface it?s in contact with, the surface itself has to be accelerating, as in the case of the floor of an elevator or an airplane. These considerations have nothing to do with acceleration along the surface. Consult your textbook for a review https://openstax.org/books/college-physics/pages/4-5-normal-tension-and-other-examples-of-forces ; https://openstax.org/books/college-physics/pages/5-1-friction] implies

(2)

The horizontal surface case corresponds to ?=0.

Analysis of Motion

As discussed in the previous lesson, the motion of the block down the incline can be analyzed in two different, equivalent ways. The first is using Newton?s Law, i.e. the relationship between force and acceleration. The second is using work and energy.

The forces along the incline are the parallel part of the gravity force, , and friction, .

(3)

We conclude that the acceleration can be expressed as). Note, the minus sign of the second term assumes the motion i.e. the velocity of the object is downward, which is exactly the case, for the case of the block sliding down inclined plane.

In the first part of the experiment, . So, in the first part of the experiment, .

In the second part of the experiment, one of the goals of this lesson will be to calculate , from measurements of velocities and positions.

We start with the familiar equation relating velocities of the object at the two photogate locations

(4)

where all these quantities are measured in the direction parallel to the incline. is the velocity at position of the second photogate. is the velocity at position of the first photogate.

We can solve for the acceleration and equate to the aforementioned expression in terms of g. Note that the expression on the left, in terms of velocities and positions, is in terms of quantities we measure. The expression on the right, comes from our understanding of the forces acting on the block.

We could now solve for the coefficient of kinetic friction, ?, from the above equation. However, our goal is not just to solve for ?. We would like to reframe the problem in terms of energy variables.

We can express the sine in terms of the ratio of height change to position change:

Now, if we simply multiply the above equation by M, we find our energy relation.

(6)

We can identify the following quantities in this equation[ If needed, consult your textbook for a review https://openstax.org/books/college-physics/pages/7-introduction-to-work-energy-and-energy-resources].

Kinetic energy:

Gravitational potential energy:

Work of friction force:

Equation (6) can then be written as

(7)

The sum of the kinetic energy and the gravitational potential energy is sometimes referred to as mechanical energy, E. We can write that the change in the mechanical energy is equal to the work done by the friction force.

(8)

Solving for ?, we obtain

(Note that there is no mass dependence in ?, as it cancels between the numerator and denominator.)

The above reformulation of the motion in terms of energy quantities, i.e. in terms of mechanical energy and work is just a matter of algebraic manipulation. However, equation is actually valid more generally. For example, potential energy can be defined not only for gravitational force, but for any conservative force. Work can be defined for any force, not only friction, and represents the amount of energy which the action of that force converts from one form into another (e.g. kinetic energy converted into heat via friction force.) Energy methods become essential, when one analyzes multi object systems.

Assignment

?Use the data in the accompanying excel file and fill out the two tables. Calculate the coefficient of kinetic friction for each part of the experiment and compare results. Both parts of the experiment are performed for three different masses. The second part, in which the block accelerates has two trials, corresponding to two different angles. You are asked to compute kinetic and gravitational potential energies, and the work done by the friction force.

?Answer the questions below.

Questions

Watch the video and fill out the excel file first. Then look at questions.

1.How is the speed of the block measured in the experiment?

2.Comment on whether and how well the two different calculations of the coefficient of kinetic friction agree.

3.We have seen that the mass cancels in the formulas for calculating the coefficients of friction. Look at your data and comment on whether you see any dependence on mass, i.e. a pattern in the differences between the values of ? for different masses.

4.Is the force of friction supposed to depend on the angle of the surface? Is the coefficient of friction supposed to depend on the angle of the surface (based on the formulas and theoretical model for friction)? Comment on whether your data shows any dependence of ? on the angle. Is there any pattern in the data?