Physics Solved Examples
Peppin Kishore
Earlier in Lesson 6, four kinematic equations were introduced and discussed. A useful problem-solving strategy was presented for use with these equations and two examples were given that illustrated the use of the strategy. Then, the application of the kinematic equations and the problem-solving strategy to free-fall motion was discussed and illustrated. In this part of Lesson 6, several sample problems will be presented. These problems allow any student of physics to test their understanding of the use of the four kinematic equations to solve problems involving the one-dimensional motion of objects. You are encouraged to read each problem and practice the use of the strategy in the solution of the problem. Then click the button to check the answer or use the link to view the solution.
These are straightforward problems that take you between two closely related concepts. Definition problems may be strictly mathematical (e.g. components of a vector), may involve rates (e.g. acceleration is the rate at which velocity changes), or they may simply be definitions (e.g. pressure is defined as force/area).
Also known as motion problems, these problems ask you to describe motion. Time is a key variable that tells you to work with the kinematic equations. If you are only asked for positions and velocities, you may also be able to work the problem using Conservation of Energy.
These problems relate speed of an object at different positions. In order to work a problem using Conservation of Energy, you need to know either that there are no significant forces taking energy out of the system or the size of those forces. Conservation of Energy will not tell you about the time it takes to go between two positions.
Electricity & Magnetism problems are often found in other categories. In addition to definition problems (e.g. electric force or field due to point charges), you use electric force in Dynamics problems and electric energy in Conservation of Energy problems. Unique to Electricity & Magnetism, however, are problems involving electric circuits or electromagnetic induction.
Two areas of modern physics are addressed through example problems on this page. Special Relativity problems ask you to relate the observations of two observers measuring the same thing. In Quantum Mechanics problems, you may look at wave or particle behavior of light and subatomic particles. As always, basic definitions problems are found with other Definitions examples.
Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail.
Two tugboats push on a barge at different angles ((Figure)). The first tugboat exerts a force of [latex] 2.7\,\,10^5\,\textN [/latex] in the x-direction, and the second tugboat exerts a force of [latex] 3.6\,\,10^5\,\textN [/latex] in the y-direction. The mass of the barge is [latex] 5.0\,\,10^6\,\textkg [/latex] and its acceleration is observed to be [latex] 7.5\,\,10^-2\,\textm/s^2 [/latex] in the direction shown. What is the drag force of the water on the barge resisting the motion? (Note: Drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object. Since the barge is flat bottomed, we can assume that the drag force is in the direction opposite of motion of the barge.)
The directions and magnitudes of acceleration and the applied forces are given in (Figure)(a). We define the total force of the tugboats on the barge as [latex] \overset\to F_\textapp [/latex] so that
The numbers used in this example are reasonable for a moderately large barge. It is certainly difficult to obtain larger accelerations with tugboats, and small speeds are desirable to avoid running the barge into the docks. Drag is relatively small for a well-designed hull at low speeds, consistent with the answer to this example, where [latex] F_\textD [/latex] is less than 1/600th of the weight of the ship.
Notice that the tension in the string is less than the weight of the block hanging from the end of it. A common error in problems like this is to set [latex] T=m_2g. [/latex] You can see from the free-body diagram of block 2 that cannot be correct if the block is accelerating.
A classic problem in physics, similar to the one we just solved, is that of the Atwood machine, which consists of a rope running over a pulley, with two objects of different mass attached. It is particularly useful in understanding the connection between force and motion. In (Figure), [latex] m_1=2.00\,\textkg [/latex] and [latex] m_2=4.00\,\textkg\text. [/latex] Consider the pulley to be frictionless. (a) If [latex] m_2 [/latex] is released, what will its acceleration be? (b) What is the tension in the string?
(The negative sign in front of [latex] m_2a [/latex] indicates that [latex] m_2 [/latex] accelerates downward; both blocks accelerate at the same rate, but in opposite directions.) Solve the two equations simultaneously (subtract them) and the result is
The result for the acceleration given in the solution can be interpreted as the ratio of the unbalanced force on the system, [latex] (m_2-m_1)g [/latex], to the total mass of the system, [latex] m_1+m_2 [/latex]. We can also use the Atwood machine to measure local gravitational field strength.
A 1.50-kg model helicopter has a velocity of [latex] 5.00\hatj\,\textm/s [/latex] at [latex] t=0. [/latex] It is accelerated at a constant rate for two seconds (2.00 s) after which it has a velocity of [latex] (6.00\hati+12.00\hatj)\textm/s\text. [/latex] What is the magnitude of the resultant force acting on the helicopter during this time interval?
A free-body diagram shows the driving force of the tractor, which gives the system its acceleration. We only need to consider motion in the horizontal direction. The vertical forces balance each other and it is not necessary to consider them. For part b, we make use of a free-body diagram of the tractor alone to determine the force between it and cart A. This exposes the coupling force [latex] \overset\to T, [/latex] which is our objective.
Since acceleration is a function of time, we can determine the velocity of the tractor by using [latex] a=\fracdvdt [/latex] with the initial condition that [latex] v_0=0 [/latex] at [latex] t=0. [/latex] We integrate from [latex] t=0 [/latex] to [latex] t=3\text: [/latex]
Since the force varies with time, we must use calculus to solve this problem. Notice how the total mass of the system was important in solving (Figure)(a), whereas only the mass of the truck (since it supplied the force) was of use in (Figure)(b).
A 10.0-kg mortar shell is fired vertically upward from the ground, with an initial velocity of 50.0 m/s (see (Figure)). Determine the maximum height it will travel if atmospheric resistance is measured as [latex] F_\textD=(0.0100v^2)\,\textN, [/latex] where v is the speed at any instant.
Initially, [latex] y_0=0 [/latex] and [latex] v_0=50.0\,\textm/s\text. [/latex] At the maximum height [latex] y=h,v=0. [/latex] The free-body diagram shows [latex] F_\textD [/latex] to act downward, because it slows the upward motion of the mortar shell. Thus, we can write
Notice the need to apply calculus since the force is not constant, which also means that acceleration is not constant. To make matters worse, the force depends on v (not t), and so we must use the trick explained prior to the example. The answer for the height indicates a lower elevation if there were air resistance. We will deal with the effects of air resistance and other drag forces in greater detail in Drag Force and Terminal Speed.
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This Special Issue of JPP is titled "Solved and Unsolved Problems in Plasma Physics". It is intended as a follow-up to the symposium that will occur in Princeton (New Jersey, USA) in March, 2016 on the occasion of Professor Nathaniel J. Fisch's 65th birthday and will honor the many innovative scientific contributions of Professor Fisch to plasma physics.
The collection of papers that will be published in this Special Issue are aimed to determine, in broad strokes, what was accomplished in fundamental theory and applications of plasma physics within the last 40 years. A unique emphasis is intended to be on open questions and unsolved problems. We hope to survey recent advances and foresee future directions in both research and education.
Coaxially nested intense $E\times B$ sheared flow realized an upgraded stable mirror plasma regime. After such an external control of high vorticity formation due to electron cyclotron heating, significantly unstable plasmas appeared. Thereby, the associated cross-field transport caused a crash of plasmas. Its generalized physics and interpretation could prepare or extend to another possibility of stability in a field-reversed configuration (FRC), for instance. Such underlying physics bases of vorticity formation were essentially or partially performed in tokamaks and stellarators (solved problems). Nevertheless, it remains to be seen whether this mirror-based experimental evidence is applicable or not to open ended FRC devices. This open issue may give a solution of one of unsolved important problems, and possibly provide more generalized and externally controllable opportunities for not only FRC but wider plasma confinement improvements.
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