Solve the following two physics problems, showing all steps and explaining your reasoning. Problem 1: A block of mass $m = 2.0$ kg is initially at rest at the top of a frictionless inclined plane that makes an angle $\theta = 30^{\circ}$ with the horizontal. The block slides down the plane a distance $d = 5.0$ m. Calculate the speed of the block at the bottom of the incline. Problem 2: Two point charges, $q_1 = +5.0 \mu C$ and $q_2 = -3.0 \mu C$, are separated by a distance $r = 0.10$ m in a vacuum. Calculate the magnitude and direction of the electric force between the charges. Discuss the implications of Coulomb's Law for this interaction.
This document presents solutions to two fundamental physics problems, one from classical mechanics and the other from electromagnetism. The aim is to illustrate a structured approach to problem-solving, emphasizing the clear application of physical principles and mathematical formulation.
Problem 1: Mechanics - Inclined Plane Dynamics
A block of mass $m = 2.0$ kg is initially at rest at the top of a frictionless inclined plane that makes an angle $\theta = 30^{\circ}$ with the horizontal. The block slides down the plane a distance $d = 5.0$ m. Calculate the speed of the block at the bottom of the incline.
1. Identify Physical Principles: This problem involves motion under gravity on an inclined plane. Since the plane is frictionless, mechanical energy is conserved. Alternatively, Newton's second law can be applied to find the acceleration and then use kinematic equations.
2. Choose a Method: Conservation of Energy Conservation of energy is often the most straightforward method for problems involving changes in height and speed, especially when friction is absent.
- Initial State (Top of incline): The block is at rest, so its initial kinetic energy ($KE_i$) is 0. Its initial potential energy ($PE_i$) is $mgh$, where $h$ is the initial height.
- Final State (Bottom of incline): The block has a final speed $v_f$, so its final kinetic energy ($KE_f$) is $\frac{1}{2}mv_f^2$. We can set the potential energy at the bottom to be 0, so $PE_f = 0$.
3. Relate Height to Distance: The height $h$ from which the block starts can be related to the distance $d$ along the incline and the angle $\theta$. From trigonometry, $h = d \sin{\theta}$.
4. Apply Conservation of Energy: The total initial mechanical energy ($E_i$) equals the total final mechanical energy ($E_f$): $E_i = KE_i + PE_i = KE_f + PE_f$ $0 + mgh = \frac{1}{2}mv_f^2 + 0$ $mgh = \frac{1}{2}mv_f^2$
5. Substitute and Solve for $v_f$: Substitute $h = d \sin{\theta}$ into the energy equation: $mg(d \sin{\theta}) = \frac{1}{2}mv_f^2$
Notice that the mass $m$ cancels out: $gd \sin{\theta} = \frac{1}{2}v_f^2$
Rearrange to solve for $v_f$: $v_f^2 = 2gd \sin{\theta}$ $v_f = \sqrt{2gd \sin{\theta}}$
6. Plug in Values and Calculate: Given $g = 9.8$ m/s$^2$, $d = 5.0$ m, and $\theta = 30^{\circ}$ ($\sin{30^{\circ}} = 0.5$): $v_f = \sqrt{2 \times 9.8 \text{ m/s}^2 \times 5.0 \text{ m} \times 0.5}$ $v_f = \sqrt{49 \text{ m}^2/ ext{s}^2}$ $v_f = 7.0$ m/s
Conclusion for Problem 1: The speed of the block at the bottom of the incline is 7.0 m/s.
Problem 2: Electromagnetism - Coulomb's Law
Two point charges, $q_1 = +5.0 \mu C$ and $q_2 = -3.0 \mu C$, are separated by a distance $r = 0.10$ m in a vacuum. Calculate the magnitude and direction of the electric force between the charges. Discuss the implications of Coulomb's Law for this interaction.
1. Identify Physical Principles: This problem concerns the electrostatic force between two point charges. Coulomb's Law quantifies this force.
2. State Coulomb's Law: Coulomb's Law states that the magnitude of the electric force ($F_e$) between two point charges ($q_1$ and $q_2$) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance ($r$) between them: $F_e = k \frac{|q_1 q_2|}{r^2}$ where $k$ is Coulomb's constant, approximately $8.98755 \times 10^9$ N m$^2$/C$^2$. For calculations, $k \approx 9.0 \times 10^9$ N m$^2$/C$^2$ is often used.
3. Convert Units: Ensure all quantities are in SI units. The charges are given in microcoulombs ($\mu C$). $q_1 = +5.0 \mu C = +5.0 \times 10^{-6}$ C $q_2 = -3.0 \mu C = -3.0 \times 10^{-6}$ C The distance is $r = 0.10$ m.
4. Calculate the Magnitude of the Force: Substitute the values into Coulomb's Law: $F_e = (9.0 \times 10^9 \text{ N m}^2/ ext{C}^2) \frac{|(+5.0 \times 10^{-6} \text{ C})(-3.0 \times 10^{-6} \text{ C})|}{(0.10 \text{ m})^2}$ $F_e = (9.0 \times 10^9) \frac{|-15.0 \times 10^{-12}|}{(0.01)}$ $F_e = (9.0 \times 10^9) \frac{15.0 \times 10^{-12}}{1.0 \times 10^{-2}}$ $F_e = (9.0 \times 10^9) \times (15.0 \times 10^{-10})$ $F_e = 135.0 \times 10^{-1}$ N $F_e = 13.5$ N
5. Determine the Direction of the Force: Since the charges have opposite signs ($q_1$ is positive, $q_2$ is negative), the force between them is attractive. The force on $q_1$ is directed towards $q_2$, and the force on $q_2$ is directed towards $q_1$. By Newton's third law, these forces are equal in magnitude and opposite in direction.
Conclusion for Problem 2: The magnitude of the electric force between the charges is 13.5 N. The force is attractive, meaning $q_1$ pulls $q_2$ towards it, and $q_2$ pulls $q_1$ towards it.
Implications of Coulomb's Law
Coulomb's Law is a cornerstone of electrostatics. Its implications include:
- Force Dependence: The force is strongly dependent on the distance between charges; it decreases rapidly as charges move apart (inverse square relationship). This explains why electrostatic forces are typically significant only over short ranges.
- Charge Interaction: The law dictates whether forces are repulsive (like charges) or attractive (opposite charges). This fundamental property governs the behavior of matter at the atomic and molecular level, influencing chemical bonding and material properties.
- Superposition Principle: While not explicitly used in this calculation, Coulomb's Law, when combined with the superposition principle, allows for the calculation of forces on charges due to multiple other charges. This is essential for understanding electric fields and potentials in complex charge distributions.
- Foundation for Electromagnetism: Coulomb's Law is a precursor to more comprehensive theories of electromagnetism, such as Maxwell's equations. It provides the basic force law from which electric fields and potentials can be derived.
Analyzing the Physics Problem Solutions
The provided solutions demonstrate a systematic approach to tackling physics problems, a skill crucial for academic success. Each problem is broken down into logical steps, making the process clear and reproducible. This structure not only aids in finding the correct answer but also in understanding the underlying physical principles.
Structure and Organization
Both solutions follow a consistent and effective structure. They begin by identifying the relevant physical principles governing the scenario. This is followed by a clear statement of the chosen method (e.g., conservation of energy, Coulomb's Law). Key steps include defining initial and final states or relevant laws, performing necessary unit conversions, applying formulas, and finally, calculating the result. Each solution concludes with a clear statement of the answer and, for the electromagnetism problem, a discussion of its implications. This organized approach ensures that no critical step is missed and that the reasoning is easy to follow.
Thesis or Claim
The implicit thesis in each solution is that applying fundamental physical laws through a step-by-step, logical process will yield a correct and interpretable result. For Problem 1, the claim is that the block's speed at the bottom can be precisely determined using energy conservation. For Problem 2, the claim is that the electrostatic force between two charges is quantifiable by Coulomb's Law, and its attractive or repulsive nature is determined by the charges' signs.
Evidence and Application of Principles
The 'evidence' in these physics problems comes directly from established physical laws and mathematical formulas. For the mechanics problem, the evidence is the principle of conservation of mechanical energy and the kinematic relationship between height and distance on an incline. For the electromagnetism problem, the evidence is Coulomb's Law itself and the definition of SI units for charge and distance. The solutions meticulously apply these principles, substituting given values into derived or stated equations. The calculations serve as the direct application of this evidence to reach the conclusion.
Tone and Language
The tone is objective, formal, and instructional. It avoids colloquialisms and focuses on precise scientific language. Mathematical notation is used correctly, and units are consistently applied. The language is direct, aiming for clarity and accuracy rather than elaborate prose. This formal tone is appropriate for academic scientific writing, ensuring that the focus remains on the physics and the solution process.
Revision Opportunities and Best Practices
While these solutions are sound, students can always refine their approach. Key areas for revision and practice include:
* Diagrams: For both problems, sketching a diagram (free-body diagram for mechanics, charge configuration for electromagnetism) at the outset can significantly clarify the setup and forces involved. This visual aid is a powerful tool for problem-solving.
* Unit Consistency: Always double-check that all quantities are in consistent SI units before calculation. Errors in unit conversion are common pitfalls.
* Significant Figures: Pay attention to significant figures in the given data and report the final answer with an appropriate number of significant figures. In these examples, values are kept to two or three significant figures.
* Conceptual Check: After obtaining a numerical answer, perform a quick conceptual check. Does the answer make physical sense? For instance, if a speed calculation results in a value faster than light, something is wrong. If a force calculation yields zero when charges are present, re-examine the setup.
* Alternative Methods: For many physics problems, multiple solution paths exist. Solving a problem using a different method (e.g., using Newton's laws for Problem 1) can serve as a valuable check and deepen understanding.
- Read the problem carefully and identify all given information and what needs to be found.
- Draw a diagram or sketch to visualize the physical situation.
- Identify the relevant physical principles and laws (e.g., Newton's laws, conservation of energy, Coulomb's Law).
- Choose an appropriate method of solution.
- List all known and unknown variables.
- Ensure all quantities are in consistent SI units.
- Write down the relevant equations.
- Solve the equations algebraically for the unknown variable before substituting numerical values.
- Substitute the numerical values and calculate the final answer.
- Check the units of the final answer.
- Perform a conceptual check to see if the answer is physically reasonable.
- State the final answer clearly, including units and appropriate significant figures.
Example: Applying the Superposition Principle
Consider three point charges: $q_A = +2.0 \mu C$ at the origin (0,0), $q_B = -3.0 \mu C$ at (0.1 m, 0), and $q_C = +4.0 \mu C$ at (0, 0.1 m). What is the net electric force on $q_A$ due to $q_B$ and $q_C$?
1. Force due to $q_B$ on $q_A$ ($F_{BA}$):
$q_A$ and $q_B$ have opposite signs, so the force is attractive. $q_B$ is to the right of $q_A$, so $F_{BA}$ is directed to the right (positive x-direction).
$r_{BA} = 0.1$ m.
$F_{BA} = k \frac{|q_A q_B|}{r_{BA}^2} = (9.0 \times 10^9) \frac{|(2.0 \times 10^{-6})(-3.0 \times 10^{-6})|}{(0.1)^2} = 18.0$ N.
So, $\vec{F}_{BA} = +18.0 \hat{i}$ N.
2. Force due to $q_C$ on $q_A$ ($F_{CA}$):
$q_A$ and $q_C$ have opposite signs, so the force is attractive. $q_C$ is above $q_A$, so $F_{CA}$ is directed upwards (positive y-direction).
$r_{CA} = 0.1$ m.
$F_{CA} = k \frac{|q_A q_C|}{r_{CA}^2} = (9.0 \times 10^9) \frac{|(2.0 \times 10^{-6})(4.0 \times 10^{-6})|}{(0.1)^2} = 36.0$ N.
So, $\vec{F}_{CA} = +36.0 \hat{j}$ N.
3. Net Force on $q_A$ ($\vec{F}_{net}$):
Using the superposition principle, $\vec{F}_{net} = \vec{F}_{BA} + \vec{F}_{CA}$.
$\vec{F}_{net} = (18.0 \hat{i} + 36.0 \hat{j})$ N.
Magnitude: $|\vec{F}_{net}| = \sqrt{(18.0)^2 + (36.0)^2} = \sqrt{324 + 1296} = \sqrt{1620} \approx 40.3$ N.
Direction: $\theta = \arctan(\frac{36.0}{18.0}) = \arctan(2) \approx 63.4^{\circ}$ above the positive x-axis.
This example shows how Coulomb's Law is extended using vector addition to find the net force in a multi-charge system.