Oscillations & Gravitation Practice Test 2/title> <link href="https://fonts.googleapis.com/css2?family=Montserrat:wght@600&family=Roboto&display=swap" rel="stylesheet" /> <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"> </script> <style> * { box-sizing: border-box; } body { margin: 0; padding: 0; font-family: 'Roboto', sans-serif; background: #f5f7fa; color: #333; line-height: 1.6; } .container { max-width: 900px; margin: 0 auto; padding: 2rem 1.5rem; background: white; border-radius: 10px; box-shadow: 0 10px 25px rgba(0, 0, 0, 0.08); } h1, h2 { font-family: 'Montserrat', sans-serif; color: #003049; } h1 { font-size: 2.6rem; margin-bottom: 1rem; text-align: center; } h2 { margin-top: 2rem; color: #0077b6; } .question { margin-bottom: 2rem; } .question p { font-size: 1.1rem; margin-bottom: 0.8rem; } .options label { display: block; margin-bottom: 0.5rem; cursor: pointer; } .question-number { font-weight: bold; color: #0077b6; margin-bottom: 0.4rem; } .feedback { display: none; margin-top: 8px; font-size: 0.95rem; color: #0077b6; } .back-link { display: inline-block; margin-bottom: 1rem; color: #005a9c; text-decoration: none; font-weight: bold; font-size: 2rem; } .back-link:hover { text-decoration: underline; transform: scale(1.02); } #submit-btn { background-color: #007bff; color: white; font-weight: 600; padding: 12px 24px; border: none; border-radius: 12px; transition: all 0.3s ease; box-shadow: 0 4px 10px rgba(0, 123, 255, 0.2); margin-top: 1rem; cursor: pointer; } #submit-btn:hover { background-color: #0056b3; transform: scale(1.05); box-shadow: 0 6px 14px rgba(0, 86, 179, 0.3); } #result { font-size: 1.2rem; margin-top: 1rem; font-weight: 600; color: #0077b6; } .back-link { display: inline-block; margin-bottom: 1rem; color: #005a9c; text-decoration: none; font-weight: bold; font-size: 2rem; } .back-link:hover { text-decoration: underline; transform: scale(1.02); } </style> </head> <body> <div class="container"> <a href="Oscillatons-Gravitation.html" class="back-link">← Back to Concepts</a> <h1>Oscillations & Gravitation Practice Test – Set B</h1> <h2>10 Questions – New Mix of Conceptual & Calculations</h2>    <div class="question"> <div class="question-number">1.</div> <p>A spring-mass system oscillates with total mechanical energy \(E\). What is the kinetic energy when the displacement is half the amplitude?</p> <div class="options"> <label><input type="radio" name="q1" value="a" /> a) \(\frac{1}{4}E\)</label> <label><input type="radio" name="q1" value="b" /> b) \(\frac{3}{4}E\)</label> <label><input type="radio" name="q1" value="c" /> c) \(\frac{1}{2}E\)</label> <label><input type="radio" name="q1" value="d" /> d) \(\frac{1}{8}E\)</label> </div> <div class="feedback" id="feedback-q1" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">2.</div> <p>A planet orbits a star in a perfect circle. What provides the centripetal force?</p> <div class="options"> <label><input type="radio" name="q2" /> a) Planet's inertia</label> <label><input type="radio" name="q2" /> b) Gravitational force</label> <label><input type="radio" name="q2" /> c) Electromagnetic force</label> <label><input type="radio" name="q2" /> d) Normal force</label> </div> <div class="feedback" id="feedback-q2" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">3.</div> <p>The time period of a mass \(m\) on a spring is \(T\). If the spring constant is doubled, what happens to the period?</p> <div class="options"> <label><input type="radio" name="q3" /> a) \(T\sqrt{2}\)</label> <label><input type="radio" name="q3" /> b) Halves</label> <label><input type="radio" name="q3" /> c) \(\frac{T}{\sqrt{2}}\) </label> <label><input type="radio" name="q3" /> d) Doubles</label> </div> <div class="feedback" id="feedback-q3" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">4.</div> <p>For a planet orbiting the Sun, what is the relationship between orbital period \(T\) and orbital radius \(r\)?</p> <div class="options"> <label><input type="radio" name="q4" /> a) \(T \propto r\)</label> <label><input type="radio" name="q4" /> b) \(T^2 \propto r\)</label> <label><input type="radio" name="q4" /> c) \(T \propto r^2\)</label> <label><input type="radio" name="q4" /> d) \(T^2 \propto r^3\)</label> </div> <div class="feedback" id="feedback-q4" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">5.</div> <p>A mass is dropped from rest and oscillates on a vertical spring. What is the equilibrium position relative to the unstretched spring?</p> <div class="options"> <label><input type="radio" name="q5" /> a) \(mg\)</label> <label><input type="radio" name="q5" /> b) \(\frac{mg}{k}\)</label> <label><input type="radio" name="q5" /> c) \(\frac{k}{mg}\)</label> <label><input type="radio" name="q5" /> d) \(\sqrt{\frac{mg}{k}}\)</label> </div> <div class="feedback" id="feedback-q5" style="display:none; margin-top:8px;"></div> </div>   <div class="question"> <div class="question-number">6.</div> <p>Which of the following increases the period of a simple pendulum?</p> <div class="options"> <label><input type="radio" name="q6" value="a" /> a) Increasing the mass</label> <label><input type="radio" name="q6" value="b" /> b) Increasing the amplitude</label> <label><input type="radio" name="q6" value="c" /> c) Increasing the length</label> <label><input type="radio" name="q6" value="d" /> d) Going to a planet with a greater gravitational acceleration</label> </div> <div class="feedback" id="feedback-q6">Period of a pendulum is \(T = 2\pi\sqrt{L/g}\), so increasing length increases period.</div> </div> <div class="question"> <div class="question-number">7.</div> <p>A 2-kg mass attached to a vertical spring stretches it by 0.2 m. What is the spring constant?</p> <div class="options"> <label><input type="radio" name="q7" value="a" /> a) 5 N/m</label> <label><input type="radio" name="q7" value="b" /> b) 10 N/m</label> <label><input type="radio" name="q7" value="c" /> c) 20 N/m</label> <label><input type="radio" name="q7" value="d" /> d) 100 N/m</label> </div> <div class="feedback" id="feedback-q7">\(F = kx \Rightarrow k = \frac{mg}{x} = \frac{2\cdot 9.8}{0.2} = 98 \approx 100 \text{ N/m}\)</div> </div> <div class="question"> <div class="question-number">8.</div> <p>What is the phase difference between displacement and acceleration in SHM?</p> <div class="options"> <label><input type="radio" name="q8" value="a" /> a) 0 radians</label> <label><input type="radio" name="q8" value="b" /> b) \(\pi/4\)</label> <label><input type="radio" name="q8" value="c" /> c) \(\pi/2\)</label> <label><input type="radio" name="q8" value="d" /> d) \(\pi\)</label> </div> <div class="feedback" id="feedback-q8">Acceleration is \(180^\circ\) or \(\pi\) radians out of phase with displacement in SHM.</div> </div> <div class="question"> <div class="question-number">9.</div> <p>What is the gravitational field strength on the surface of a planet of mass \(M\) and radius \(R\)?</p> <div class="options"> <label><input type="radio" name="q9" value="a" /> a) \(\frac{GM}{R}\)</label> <label><input type="radio" name="q9" value="b" /> b) \(\frac{GM}{R^2}\)</label> <label><input type="radio" name="q9" value="c" /> c) \(\frac{G}{MR^2}\)</label> <label><input type="radio" name="q9" value="d" /> d) \(\frac{M}{GR^2}\)</label> </div> <div class="feedback" id="feedback-q9">Field strength \(g = \frac{GM}{R^2}\), derived from Newton’s Law.</div> </div> <div class="question"> <div class="question-number">10.</div> <p>How does the total energy in SHM change as the system oscillates?</p> <div class="options"> <label><input type="radio" name="q10" value="a" /> a) It increases</label> <label><input type="radio" name="q10" value="b" /> b) It decreases</label> <label><input type="radio" name="q10" value="c" /> c) It stays constant</label> <label><input type="radio" name="q10" value="d" /> d) It oscillates with position</label> </div> <div class="feedback" id="feedback-q10">In ideal SHM, energy is conserved: total energy stays constant, only form changes between KE and PE.</div> </div> <div class="question"> <div class="question-number">11.</div> <p>What is the maximum speed \(v_{max}\) of a mass-spring oscillator with amplitude \(A\) and angular frequency \(\omega\)?</p> <div class="options"> <label><input type="radio" name="q11" /> a) \(A / \omega\)</label> <label><input type="radio" name="q11" /> b) \(\omega A\)</label> <label><input type="radio" name="q11" /> c) \(A \omega^2\)</label> <label><input type="radio" name="q11" /> d) \(1 / (\omega A)\)</label> </div> <div class="feedback" id="feedback-q11" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">12.</div> <p>In gravitational potential energy \(U = -\frac{GMm}{r}\), what does the negative sign indicate?</p> <div class="options"> <label><input type="radio" name="q12" /> a) Energy increases with decreasing \(r\)</label> <label><input type="radio" name="q12" /> b) Energy is always positive</label> <label><input type="radio" name="q12" /> c) Work must be done to separate masses to infinity</label> <label><input type="radio" name="q12" /> d) No work is needed to bring masses together</label> </div> <div class="feedback" id="feedback-q12" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">13.</div> <p>A mass is oscillating with frequency \(f\). What happens to the period if frequency doubles?</p> <div class="options"> <label><input type="radio" name="q13" /> a) Period doubles</label> <label><input type="radio" name="q13" /> b) Period halves</label> <label><input type="radio" name="q13" /> c) Period quadruples</label> <label><input type="radio" name="q13" /> d) Period stays the same</label> </div> <div class="feedback" id="feedback-q13" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">14.</div> <p>The acceleration of a mass in SHM is given by \(a = -\omega^2 x\). What is the significance of the negative sign?</p> <div class="options"> <label><input type="radio" name="q14" /> a) Acceleration is always negative</label> <label><input type="radio" name="q14" /> b) Acceleration opposes displacement</label> <label><input type="radio" name="q14" /> c) Velocity decreases over time</label> <label><input type="radio" name="q14" /> d) Displacement is negative</label> </div> <div class="feedback" id="feedback-q14" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">15.</div> <p>A satellite orbits Earth in a circular orbit. What happens to orbital speed if the radius doubles?</p> <div class="options"> <label><input type="radio" name="q15" /> a) Speed doubles</label> <label><input type="radio" name="q15" /> b) Speed halves</label> <label><input type="radio" name="q15" /> c) Speed decreases by factor \(1/\sqrt{2}\)</label> <label><input type="radio" name="q15" /> d) Speed decreases by factor \(1/\sqrt{2}\)</label> </div> <div class="feedback" id="feedback-q15" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">16.</div> <p>For a mass \(m\) on a spring, if the amplitude is doubled, what happens to the total mechanical energy?</p> <div class="options"> <label><input type="radio" name="q16" /> a) It doubles</label> <label><input type="radio" name="q16" /> b) It quadruples</label> <label><input type="radio" name="q16" /> c) It halves</label> <label><input type="radio" name="q16" /> d) It stays the same</label> </div> <div class="feedback" id="feedback-q16" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">17.</div> <p>Which expression gives the gravitational potential energy of a two-mass system separated by distance \(r\)?</p> <div class="options"> <label><input type="radio" name="q17" /> a) \(\frac{GMm}{r}\)</label> <label><input type="radio" name="q17" /> b) \(-\frac{GMm}{r}\)</label> <label><input type="radio" name="q17" /> c) \(\frac{GM}{r^2}\)</label> <label><input type="radio" name="q17" /> d) \(-\frac{GM}{r^2}\)</label> </div> <div class="feedback" id="feedback-q17" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">18.</div> <p>If a pendulum clock is taken to the Moon (where \(g\) is smaller), what happens to its period?</p> <div class="options"> <label><input type="radio" name="q18" /> a) Period increases</label> <label><input type="radio" name="q18" /> b) Period decreases</label> <label><input type="radio" name="q18" /> c) Period remains unchanged</label> <label><input type="radio" name="q18" /> d) Period becomes zero</label> </div> <div class="feedback" id="feedback-q18" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">19.</div> <p>The restoring force in SHM is proportional to:</p> <div class="options"> <label><input type="radio" name="q19" /> a) Velocity</label> <label><input type="radio" name="q19" /> b) Acceleration</label> <label><input type="radio" name="q19" /> c) Displacement</label> <label><input type="radio" name="q19" /> d) Mass</label> </div> <div class="feedback" id="feedback-q19" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">20.</div> <p>If the mass in a mass-spring system is quadrupled, what happens to the period?</p> <div class="options"> <label><input type="radio" name="q20" /> a) Period doubles</label> <label><input type="radio" name="q20" /> b) Period quadruples</label> <label><input type="radio" name="q20" /> c) Period halves</label> <label><input type="radio" name="q20" /> d) Period stays the same</label> </div> <div class="feedback" id="feedback-q20" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">21.</div> <p>A 0.5 kg mass attached to a spring with \(k = 200\,N/m\) oscillates with amplitude 0.1 m. What is the maximum speed?</p> <div class="options"> <label><input type="radio" name="q21" /> a) 2 m/s</label> <label><input type="radio" name="q21" /> b) 1 m/s</label> <label><input type="radio" name="q21" /> c) 0.5 m/s</label> <label><input type="radio" name="q21" /> d) 4 m/s</label> </div> <div class="feedback" id="feedback-q21" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">22.</div> <p>A pendulum of length 1.5 m swings on Earth (\(g = 9.8\,m/s^2\)). What is the period?</p> <div class="options"> <label><input type="radio" name="q22" /> a) 1.55 s</label> <label><input type="radio" name="q22" /> b) 2.45 s</label> <label><input type="radio" name="q22" /> c) 2.45 s</label> <label><input type="radio" name="q22" /> d) 3.10 s</label> </div> <div class="feedback" id="feedback-q22" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">23.</div> <p>A planet has mass \(6 \times 10^{24}\) kg and radius \(6.4 \times 10^6\) m. What is the gravitational acceleration at its surface?</p> <div class="options"> <label><input type="radio" name="q23" /> a) 5.0 m/s²</label> <label><input type="radio" name="q23" /> b) 8.0 m/s²</label> <label><input type="radio" name="q23" /> c) 9.8 m/s²</label> <label><input type="radio" name="q23" /> d) 12.0 m/s²</label> </div> <div class="feedback" id="feedback-q23" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">24.</div> <p>A mass-spring oscillator has amplitude 0.3 m and angular frequency 5 rad/s. What is the maximum acceleration?</p> <div class="options"> <label><input type="radio" name="q24" /> a) 1.5 m/s²</label> <label><input type="radio" name="q24" /> b) 5 m/s²</label> <label><input type="radio" name="q24" /> c) 7.5 m/s²</label> <label><input type="radio" name="q24" /> d) 7.5 m/s²</label> </div> <div class="feedback" id="feedback-q24" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">25.</div> <p>An object in circular orbit has orbital radius \(2 \times 10^7\) m and speed \(3 \times 10^3\) m/s. What is the centripetal acceleration?</p> <div class="options"> <label><input type="radio" name="q25" /> a) 0.15 m/s²</label> <label><input type="radio" name="q25" /> b) 0.45 m/s²</label> <label><input type="radio" name="q25" /> c) 0.90 m/s²</label> <label><input type="radio" name="q25" /> d) 1.5 m/s²</label> </div> <div class="feedback" id="feedback-q25" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">26.</div> <p>A 3 kg mass attached to a horizontal spring oscillates with frequency 2 Hz. What is the spring constant?</p> <div class="options"> <label><input type="radio" name="q26" /> a) 150 N/m</label> <label><input type="radio" name="q26" /> b) 474 N/m</label> <label><input type="radio" name="q26" /> c) 947 N/m</label> <label><input type="radio" name="q26" /> d) 1894 N/m</label> </div> <div class="feedback" id="feedback-q26" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">27.</div> <p>The potential energy of a spring compressed by 0.05 m with \(k = 300 N/m\) is:</p> <div class="options"> <label><input type="radio" name="q27" /> a) 0.375 J</label> <label><input type="radio" name="q27" /> b) 0.5 J</label> <label><input type="radio" name="q27" /> c) 1.25 J</label> <label><input type="radio" name="q27" /> d) 3.75 J</label> </div> <div class="feedback" id="feedback-q27" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">28.</div> <p>An object of mass 5 kg oscillates with amplitude 0.2 m and frequency 1 Hz. What is the total energy?</p> <div class="options"> <label><input type="radio" name="q28" /> a) 1.97 J</label> <label><input type="radio" name="q28" /> b) 3.95 J</label> <label><input type="radio" name="q28" /> c) 7.9 J</label> <label><input type="radio" name="q28" /> d) 15.8 J</label> </div> <div class="feedback" id="feedback-q28" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">29.</div> <p>If the period of a mass-spring system is 0.5 s, what is its angular frequency?</p> <div class="options"> <label><input type="radio" name="q29" /> a) \(\pi\) rad/s</label> <label><input type="radio" name="q29" /> b) \(2\pi\) rad/s</label> <label><input type="radio" name="q29" /> c) \(4\pi\) rad/s</label> <label><input type="radio" name="q29" /> d) \(8\pi\) rad/s</label> </div> <div class="feedback" id="feedback-q29" style="display:none; margin-top:8px;"></div> </div> <div class="question"> <div class="question-number">30.</div> <p>A planet of mass \(6 \times 10^{24}\) kg orbits at radius \(4.2 \times 10^7\) m. What is the gravitational potential energy of a 1 kg satellite there?</p> <div class="options"> <label><input type="radio" name="q30" /> a) \(-1.0 \times 10^8\) J</label> <label><input type="radio" name="q30" /> b) \(-9.5 \times 10^8\) J</label> <label><input type="radio" name="q30" /> c) \(-1.0 \times 10^9\) J</label> <label><input type="radio" name="q30" /> d) \(-9.5 \times 10^9\) J</label> </div> <div class="feedback" id="feedback-q30" style="display:none; margin-top:8px;"></div> </div> <button id="submit-btn" onclick="checkAnswers()">Submit Quiz</button> <div id="result"></div> </div> <script> const answers = { q1: "b", q2: "b", q3: "c", q4: "d", q5: "b", q6: "c", q7: "d", q8: "d", q9: "b", q10: "c", q11: "b", q12: "c", q13: "b", q14: "b", q15: "c", q16: "b", q17: "b", q18: "a", q19: "c", q20: "a", q21: "a", q22: "c", q23: "c", q24: "d", q25: "b", q26: "b", q27: "a", q28: "b", q29: "c", q30: "b" }; const feedback= { q1: "The total mechanical energy of the system can be given as \\( \\frac{1}{2}kA^2 \\). When the displacement is \\( \\frac{A}{2} \\), the elastic potential energy is \\( \\frac{1}{2}k\\left(\\frac{A}{2}\\right)^2 = \\frac{1}{8}kA^2 \\). This is one fourth of the total mechanical energy in the system. Using conservation of energy, the other three fourths come from kinetic energy.", q2: "The gravitational force points toward the center (the star) and provides the centripetal acceleration.", q3: "Using the formula \\( T = 2\\pi \\sqrt{\\frac{m}{k}} \\), if the spring constant \\(k\\) is doubled, the period becomes \\( T = 2\\pi \\sqrt{\\frac{m}{2k}} = \\frac{1}{\\sqrt{2}} T \\).", q4: "According to Kepler's third law, \\( T^2 \\propto r^3 \\), so the square of the orbital period is proportional to the cube of the radius.", q5: "At equilibrium, the net force is zero: \\( mg = kx \\). Solving for \\(x\\) gives \\( x = \\frac{mg}{k} \\).", q6: "Using the formula \\( T = 2\\pi \\sqrt{\\frac{l}{g}} \\), increasing \\(l\\) increases the period.", q7: "At equilibrium, \\( mg = kx \\). Solving: \\( k = \\frac{mg}{x} = \\frac{2 \\times 9.8}{0.2} = 98 \\approx 100\\,\\text{N/m} \\).", q8: "In simple harmonic motion, the acceleration is always \\( \\pi \\) radians (180°) out of phase with the displacement, meaning it points in the opposite direction.", q9: "Setting gravitational force \\( F = \\frac{GMm}{R^2} \\) equal to \\(mg\\) and solving for \\(g\\) gives \\( g = \\frac{GM}{R^2} \\).", q10: "In SHM, the total energy remains constant and continuously transforms between kinetic and potential energy.", q11: "Maximum speed in SHM is given by \\(v_{max} = \\omega A\\), where \\(\\omega\\) is angular frequency and \\(A\\) is amplitude; this represents the point where all energy is kinetic.", q12: "The negative sign in gravitational potential energy \\(U = -\\frac{GMm}{r}\\) shows that work must be done to move masses from separation \\(r\\) to infinity, reflecting the attractive nature of gravity.", q13: "Frequency and period are inversely related: \\(T = 1/f\\). If frequency doubles, period halves.", q14: "In SHM, acceleration \\(a = -\\omega^2 x\\) always opposes displacement, restoring the mass toward equilibrium (the negative sign shows direction is opposite to \\(x\\)).", q15: "Orbital speed is \\(v = \\sqrt{\\frac{GM}{r}}\\). Doubling radius means \\(v\\) decreases by \\(1/\\sqrt{2}\\) (since \\(v \\propto 1/\\sqrt{r}\\)).", q16: "Total mechanical energy in SHM is \\(E = \\frac{1}{2}kA^2\\). If amplitude doubles, energy increases by factor 4 because \\(A^2\\) quadruples.", q17: "Gravitational potential energy between two masses is \\(U = -\\frac{GMm}{r}\\), negative because gravitational force is attractive.", q18: "On the Moon, \\(g\\) is smaller, so \\(T = 2\\pi\\sqrt{\\frac{L}{g}}\\) increases; thus, the period increases.", q19: "The restoring force is proportional to displacement: \\(F = -kx\\), the defining feature of SHM.", q20: "Period of a mass-spring oscillator is \\(T = 2\\pi\\sqrt{m/k}\\). Quadrupling \\(m\\) makes \\(T\\) double since \\(\\sqrt{4} = 2\\).", q21: "The maximum speed in SHM is given by \\(v_{max} = \\omega A\\), where \\(\\omega = \\sqrt{\\frac{k}{m}} = \\sqrt{\\frac{200}{0.5}} = 20\\,rad/s\\). So \\(v_{max} = 20 \\times 0.1 = 2\\,m/s\\).", q22: "The period of a pendulum is \\(T = 2\\pi \\sqrt{\\frac{L}{g}} = 2\\pi \\sqrt{\\frac{1.5}{9.8}} \\approx 2.45\\,s\\).", q23: "Using \\(g = \\frac{GM}{R^2}\\), \\(G = 6.67 \\times 10^{-11}\\). Substituting: \\(g = \\frac{6.67e-11 * 6e24}{(6.4e6)^2} \\approx 9.8\\,m/s^2\\).", q24: "Maximum acceleration is \\(a_{max} = \\omega^2 A = 5^2 * 0.3 = 25 *0.3 =7.5\\,m/s^2\\).", q25: "Centripetal acceleration: \\(a = \\frac{v^2}{r} = \\frac{(3e3)^2}{2e7} = \\frac{9e6}{2e7} =0.45\\,m/s^2\\).", q26: "Frequency \\(f=2Hz\\), \\(\\omega=2\\pi f=4\\pi\\). Then \\(k=m \\omega^2=3*(4\\pi)^2=3*16\\pi^2 \\approx 474\\,N/m\\).", q27: "Potential energy: \\(U=\\frac{1}{2}kx^2=0.5*300*0.0025=0.375J\\).", q28: "Energy: \\(E=\\frac{1}{2}kA^2\\). First, \\(\\omega=2\\pi f=2\\pi\\), \\(k=m \\omega^2=5*(2\\pi)^2=5*4\\pi^2\\approx197\\). \\(E=0.5*197*0.04\\approx3.95J\\).", q29: "Angular frequency: \\(\\omega=\\frac{2\\pi}{T}=\\frac{2\\pi}{0.5}=4\\pi\\,rad/s\\).", q30: "Potential energy: \\(U=-\\frac{GMm}{r}\\). \\(G=6.67e-11\\), \\(M=6e24\\), \\(r=4.2e7\\). \\(U=-(6.67e-11*6e24)/4.2e7 = -9.5e8J\\)." } function checkAnswers() { let score = 0; for (let i = 1; i <= 30; i++) { const questionName = "q" + i; const options = document.getElementsByName(questionName); let selected = ""; options.forEach(opt => { opt.parentElement.style.color = "black"; // Reset color if (opt.checked) selected = opt.value; }); const feedbackEl = document.getElementById("feedback-" + questionName); if (selected === answers[questionName]) { score++; options.forEach(opt => { if (opt.value === selected) opt.parentElement.style.color = "green"; }); } else { options.forEach(opt => { if (opt.value === selected) opt.parentElement.style.color = "red"; if (opt.value === answers[questionName]) opt.parentElement.style.color = "green"; }); } // Show feedback if (feedbackEl) { feedbackEl.innerText = "Explanation: " + (feedback[questionName] || "No feedback available."); feedbackEl.style.display = "block"; } if (window.MathJax) { MathJax.typeset(); // Re-render MathJax math in dynamically added content } } document.getElementById("result").innerText = `You scored ${score} out of 30.`; } // Attach values a-d to radio buttons window.onload = () => { for (let i = 1; i <= 30; i++) { const options = document.getElementsByName("q" + i); const letters = ["a", "b", "c", "d"]; options.forEach((opt, idx) => { opt.value = letters[idx]; }); } }; </script> </body> </html>