Gravity Archives - Page 5 of 5

Gravitation

Posted on December 6, 2011 by admin — No Comments ↓

Agenda:
   Interactive Physics: Gravity
   Discussion: Gravitation, Grav. Fields, Orbits
   Practice UCM & Grav Questions
HW:
   Watch video: "Gravitational Fields" & answer ?s by Wednesday

Universal Gravitation

All objects that have mass attract each other with a gravitational force. The magnitude of that force, Fg, can be calculated using Newton’s Law of Universal Gravitation:

This law tells us that the force of gravity between two objects is proportional to each of the masses(m1 and m2) and inversely proportional to the square of the distance between them (r). The universal gravitational constant, G, is a "fudge factor," so to speak, included in the equation so that your answers come out in S.I. units. G is given on the front page of your Regents Physics Reference Table as .

Let’s look at this relationship in a bit more detail. Force is directly proportional to the masses of the two objects, therefore if either of the masses were doubled, the gravitational force would also double. In similar fashion, if the distance between the two objects, r, was doubled, the force of gravity would be quartered since the distance is squared in the denominator. This type of relationship is called an inverse square law, which describes many phenomena in the natural world.

NOTE: the distance between the masses, r, is actually the distance between the center of masses of the objects. For large objects, such as the Earth, for example, you must determine the distance to the center of the Earth, not to its surface.

Some hints for problem solving when dealing with Newton’s Law of Universal Gravitation:

Substitute values in for variables at the very end of the problem only. The longer you can keep the formula in terms of variables, the fewer opportunities for mistakes.
Before using your calculator to find an answer, try to estimate the order of magnitude of the answer. Use this to check your final answer.
Once your calculations are complete, make sure your answer makes sense by comparing your answer to a known or similar quantity. If your answer doesn’t make sense, check your work and verify your calculations.

Let’s see if we can’t apply Newton’s Law of Universal Gravitation to a simple problem…

Question: What is the gravitational force of attraction between two asteroids in space, each with a mass of 50,000 kg, separated by a distance of 3800 m?

Answer:

As you can see, the force of gravity is a relatively weak force, and we would expect a relatively weak force between relatively small objects. It takes tremendous masses and small distances in order to develop significant gravitational forces. Let’s take a look at another problem to explore the relationships between gravitational force, mass, and distance.

Gravitational Fields

Gravity is a non-contact, or field, force. Its effects are observed without the two objects coming into contact with each other. Exactly how this happens is a mystery to this day, but scientists have come up with a mental construct to help us understand how gravity works.

gravfield

Envision an object with a gravitational field, such as the planet Earth. The closer other masses are to Earth, the more gravitational force they will experience. We can characterize this by calculating the amount of force the Earth will exert per unit mass at various distances from the Earth. Obviously, the closer the object is to the Earth, the larger a force it will experience, and the farther it is from the Earth, the smaller a force it will experience.

Attempting to visualize this, picture the strength of the gravitational force on a test object represented by a vector at the position of the object. The denser the force vectors are, the stronger the force, the stronger the "gravitational field." As these field lines become less and less dense, the gravitational field gets weaker and weaker.

To calculate the gravitational field strength at a given position, we can go back to our definition of the force of gravity on our test object, better known as its weight. We’ve been writing this as mg since we began our study of Dynamics. But, realizing that this is the force of gravity on an object, we can also calculate the force of gravity on test mass using Newton’s Law of Universal Gravitation. Putting these together we find that:

Realizing that the mass on the left-hand side of the equation, the mass of our test object, is also one of the masses on the right-hand side of the equation, we can simplify our expression by dividing out the test mass.

Therefore, the gravitational field strength, g, is equal to the universal gravitational constant, G, times the mass of the object, divided by the square of the distance between the objects.

But wait, you might say… I thought g was the acceleration due to gravity on the surface of the Earth! And you would be right. Not only is g the gravitational field strength, it’s also the acceleration due to gravity. The units even work out… the units of gravitational field strength, N/kg, are equivalent to the units for acceleration, m/s²!

Still skeptical? Let’s calculate the gravitational field strength on the surface of the Earth, using the knowledge that the mass of the Earth is approximately 5.98*10²⁴ kg, and the distance from the surface to the center of mass of the Earth (which varies slightly since the Earth isn’t a perfect sphere) is approximately 6378 km in New York.

derive-g

As expected, the gravitational field strength on the surface of the Earth is the acceleration due to gravity. Let’s see if we can’t solve some problems using gravitational field strength.

Question: Suppose a 100-kg astronaut feels a gravitational force of 700N when placed in the gravitational field of a planet.

A) What is the gravitational field strength at the location of the astronaut?

B) What is the mass of the planet if the astronaut is 2*10⁶ m from its center?

Answer:

Regents Physics SBG Objective Tracking Sheets

Posted on August 17, 2011 by admin — No Comments ↓

10 Quick Tips to Maximize your Regents Physics Score

Posted on June 13, 2011 by admin — No Comments ↓

Although by no means an exhaustive list, these 10 quick tips may help you secure that extra point or two on your upcoming Regents Physics exam.

Mass and inertia are the same thing.
To find the resultant, line your vectors up tip-to-tail, and draw a line from the starting point of the first vector to the ending point of the last vector.
Any object moving in a circular path is accelerating toward the center of the circle.
Acceleration of an object is equal to the net force on the object divided by the object’s mass.
The normal force always points at an angle of 90° from the surface.
Opposite charges and magnetic poles attract, likes repel.
Gravitational forces and electrostatic forces both follow an inverse square law relationship, where the strength of the force is related to one divided by the square of the distance between the charges/masses.
The force of gravity on an object, commonly referred to as weight, is equal to mg, where g is the gravitational field strength (also referred to as the acceleration due to gravity).
The mass-energy equivalence can be calculated using E=mc^2. If a mass is given in universal mass units, however, you can do a straight unit conversion using 1u = 931 MeV.
Protons and neutrons fall into the category of baryons, which are hadrons. Smaller particles, such as electrons, fall into the category of leptons. Mesons are rare, weird particles you probably haven’t heard of.

Most importantly, use your reference table. When in doubt, write down the information you’re asked to find, what you’re given, and use your reference table to help you narrow down what you should be doing. In the free response part of the test, make sure to show your work in detail with a formula, substitution with units, and an answer with units.

Find these and many more tips for success at APlusPhysics.com.

Mid-Term Exam Results 2010-2011

Posted on February 4, 2011 by admin — No Comments ↓

Mid-Term results have been compiled and published. In general, scores are right in line with performance within the last few years. The exam itself was styled after the Regents Exam which is given at the end of the school year, and included topics from all aspects of Newtonian Mechanics (roughly half of the Regents Exam). This included 46 multiple choice questions, and 25 open-ended, or “free response,” type questions. Students were given 180 minutes to complete the exam, as well as a formula sheet (Regents Physics Reference Table), and a ruler. Students were permitted the use of a scientific calculator.

Summary Data

Summary data from this year’s exam, along with the previous two years of data, are shown at right. Average score on the exam, with a fairly generous curve, was 76%, compared to 73% in 2010 and 74% in 2009. Median score was 78%, two points higher than last year’s exam, and right in line with 2009 scores. Percentage of students scoring above 85%, considered “Mastery Level” by Regents Physics standards, was 25%, slightly lower than in 2010 and 2009, but passing percentage was 83%, several points higher than in past years. Even more promising is a glance at a histogram breaking down student scores further. This data indicates that we have many students in the 80-85% range. Past history has shown me that it is quite possible for students to improve their scores from 10-15% points between now and the actual Regents Exam in June with focused effort, study, and motivation. This indicates the potential for outstanding standardized exam performance in June if students buckle down and focus their efforts. Unfortunately, the Physics Regents Exam is usually one of the last tests offered (sometimes even after official graduation), so obtaining and maintaining student focus at this point of the year can be a challenge.

Cluster Data

Breaking down exam performance by topic, we observe students demonstrating a very strong understanding of Newton’s Laws of Motion, Hooke’s Law (springs), and Work and Energy. These have been focus areas during the year, although there is room for further improvement in understanding Newton’s 2nd Law as well as momentum and impulse.

Cluster data indicates weaknesses in Newton’s Law of Universal Gravitation, although a deeper look at the questions and errors themselves indicates that most of the confusion here is a result of mathematical errors rather than conceptual misunderstandings. This is being re-emphasized as we work through Coulomb’s Law / Electrical Force problems, which utilize almost the exact same mathematical model. Other areas for improvement include 2-D Kinematics (projectile motion), a perennial challenge, and utilizing and understanding the metric system, a topic that has been emphasized and will continue to be reviewed throughout the year. Performance on momentum problems was, unsurprisingly, problematic. This relates directly to a “class as a whole” issue with independent practice and motivation throughout that unit, and will be our top priority for review in mid-June.

Conclusions

Student performance is in line with, if not slightly above, expectations for this point in the year. With a focused effort, students can fairly routinely increase their scores 10 to 15 percentage points by June. For struggling students, a variety of remediation protocols are being put in place, including the entire course content and sample quizzes available on the web, tailored directly to this course’s requirements (APlusPhysics.com), video lectures on topics corresponding directly to our textbook (Hippocampus), and even review sessions which can be viewed from any computer supporting iTunes (Windows, Mac, Linux, etc.) as well as iPods, iPhones, Blackberries, Android devices, etc. All of these resources are available both in the classroom as well as the school library for those who don’t have ready access to these resources outside of school.

Data has also been broken down to the same level for individual students, and will be utilized to develop both entire-class and individual review plans to allow students to focus on areas that will provide the biggest “bang for the buck” in their studies.