A 0.1 kg ball is thrown vertically upwards with a speed of 15 ms^-1. At the same instant, another 0.2 kg ball is dropped from a height of 20 m above the first ball’s starting point. At what height do the two balls meet? (g = 10 ms^-2)
For the first ball, height h_1 = ut - (1/2)gt^2 = 15t - 5t^2. For the second ball, height h_2 = 20 - (1/2)gt^2 = 20 - 5t^2. They meet when h_1 = h_2: 15t - 5t^2 = 20 - 5t^2 → 15t = 20 → t = 4/3 s. Height h_1 = 15 × (4/3) - 5 × (4/3)^2 = 20 - 5 × (16/9) ≈ 20 - 8.89 ≈ 9 m.
A 3 kg mass is suspended by two strings making angles of 30° and 60° with the horizontal. What is the tension in the string at 30° to the horizontal? (g = 10 ms^-2)
Vertical equilibrium: T_1 sin30° + T_2 sin60° = mg = 3 × 10 = 30 N. Horizontal equilibrium: T_1 cos30° = T_2 cos60°. Thus, T_1 (√3/2) = T_2 (1/2) → T_2 = T_1 √3. Substitute: (T_1/2) + (T_1 √3)(√3/2) = 30 → T_1/2 + 3T_1/2 = 30 → 2T_1 = 30 → T_1 = 30 N.
A 0.5 kg mass on a frictionless surface is connected to two springs of constants 100 Nm^-1 and 200 Nm^-1 in parallel. What is the period of oscillation of the system?
Equivalent spring constant k_eq = k_1 + k_2 = 100 + 200 = 300 Nm^-1. Period T = 2π√(m/k) = 2π√(0.5/300) ≈ 2π × 0.0408 ≈ 0.25 s.
A 2 kg mass moving at 4 ms^-1 collides with a 3 kg mass moving at 2 ms^-1 in the opposite direction. If the collision is perfectly elastic, what is the velocity of the 2 kg mass after the collision?
Conservation of momentum: 2 × 4 + 3 × (-2) = 2v_1 + 3v_2 → 2 = 2v_1 + 3v_2. Relative velocity for elastic collision: v_1 - v_2 = -[4 - (-2)] = -6. Solve: v_1 = -2 ms^-1, indicating the 2 kg mass moves in the opposite direction.
A 0.2 kg block is placed on a rough incline at 45°. If the coefficient of friction is 0.5, what is the minimum force parallel to the incline required to move the block up the incline? (g = 10 ms^-2)
Normal force N = mg cos45° = 0.2 × 10 × (1/√2) ≈ 1.41 N. Frictional force f = μN = 0.5 × 1.41 ≈ 0.71 N. Weight component down the incline = mg sin45° ≈ 1.41 N. Minimum force to move up = 1.41 + 0.71 + 2 × 0.2 × sin45° ≈ 4.24 N, assuming impending motion.
A 0.1 kg object floats in a liquid with 40% of its volume submerged. The liquid is then replaced by another with a density of 1200 kgm^-3, causing 60% submersion. What is the density of the first liquid? (g = 10 ms^-2)
Object density ρ_o = fraction submerged × liquid density. For the second liquid: ρ_o = 0.6 × 1200 = 720 kgm^-3. For the first liquid: 720 = 0.4 × ρ_1 → ρ_1 = 720 / 0.4 = 1500 kgm^-3.
A gas expands isothermally from 0.1 m^3 to 0.3 m^3 at a temperature of 300 K. If the initial pressure is 2 x 10^5 Pa, how much work is done by the gas?
Work done W = nRT ln(V_2/V_1). From P_1V_1 = nRT, nRT = 2 × 10^5 × 0.1 = 2 × 10^4 J. Thus, W = 2 × 10^4 × ln(0.3/0.1) = 2 × 10^4 × ln(3) ≈ 3.46 × 10^4 J.
A 0.4 kg piece of ice at 0°C is placed in 0.6 kg of water at 50°C. What is the final temperature of the mixture? (Specific heat of water = 4200 Jkg^-1K^-1, latent heat of fusion of ice = 3.34 x 10^5 Jkg^-1)
Heat to melt ice = mL = 0.4 × 3.34 × 10^5 = 1.336 × 10^5 J. Heat lost by water to 0°C = 0.6 × 4200 × 50 = 1.26 × 10^5 J. Remaining heat = 1.336 × 10^5 - 1.26 × 10^5 = 7600 J. Final temperature: (0.4 + 0.6) × 4200 × T = 7600 → T ≈ 20°C.
A heat engine operates with an efficiency of 40% and produces 5000 J of work per cycle. If the heat input occurs at 600 K, what is the temperature of the cold reservoir?
Efficiency η = W / Q_h → 0.4 = 5000 / Q_h → Q_h = 12,500 J. Carnot efficiency η = 1 - (T_c / T_h) → 0.4 = 1 - (T_c / 600) → T_c = 600 × 0.6 = 360 K.
A transverse wave on a string has an amplitude of 0.05 m and a frequency of 200 Hz. If the maximum transverse acceleration of a particle on the string is 8000 ms^-2, what is the speed of the wave?
Maximum acceleration a_max = Aω^2 = A (2πf)^2. Given a_max = 8000, A = 0.05, f = 200: 8000 = 0.05 × (2π × 200)^2 → ω = 400π. Wave speed v = ω / k, where k = ω / v. Using a_max = Aω^2, derive v = √(a_max / (A × 4π^2f^2)) × (2πf) ≈ 25 ms^-1.
A stationary wave is set up on a string of length 1.2 m fixed at both ends. If the speed of the wave is 480 ms^-1, what is the frequency of the 4th harmonic?
Fundamental frequency f_1 = v / (2L) = 480 / (2 × 1.2) = 200 Hz. The 4th harmonic f_4 = 4 × f_1 = 4 × 200 = 800 Hz.
A diffraction grating with 500 lines per mm is illuminated with light of wavelength 600 nm. What is the angle of the second-order maximum?
Grating spacing d = 1 / (500 × 10^3) = 2 × 10^-6 m. For second order (m = 2): d sinθ = mλ → (2 × 10^-6) sinθ = 2 × (600 × 10^-9) → sinθ = 0.6 → θ = sin^-1(0.6) ≈ 36.9°.
A telescope has an objective lens of focal length 100 cm and an eyepiece of focal length 5 cm. If an object is placed at infinity, what is the angular magnification of the telescope?
Angular magnification M = f_o / f_e = 100 / 5 = 20, representing the ratio of the objective’s focal length to the eyepiece’s focal length.
A proton moves at 5 x 10^6 ms^-1 perpendicular to a magnetic field of 0.2 T. What is the radius of its circular path? (m_p = 1.67 x 10^-27 kg, q = 1.6 x 10^-19 C)
Radius r = mv / (qB) = (1.67 × 10^-27 × 5 × 10^6) / (1.6 × 10^-19 × 0.2) ≈ 0.26 m, derived from the balance of magnetic and centripetal forces.
A 10 μF capacitor and a 20 μF capacitor are connected in series to a 24 V battery. What is the potential difference across the 10 μF capacitor?
Equivalent capacitance C_eq = (10 × 20) / (10 + 20) = 20/3 μF. Charge Q = C_eq × V = (20/3) × 10^-6 × 24 = 160 × 10^-6 C. Voltage across 10 μF = Q / C = (160 × 10^-6) / (10 × 10^-6) = 16 V.
A 12 V battery with an internal resistance of 2 Ω is connected to a load resistor. If the maximum power is delivered to the load, what is the value of the load resistor?
Maximum power is delivered when the load resistance equals the internal resistance, so R_load = 2 Ω, as per the maximum power transfer theorem.
A transformer has an efficiency of 90% and steps up the voltage from 200 V to 1000 V. If the output power is 4500 W, what is the primary current?
Efficiency η = P_out / P_in → 0.9 = 4500 / P_in → P_in = 4500 / 0.9 = 5000 W. Primary current I_p = P_in / V_p = 5000 / 200 = 25 A.
A radioactive sample has an initial activity of 800 counts per minute. After 12 hours, the activity drops to 200 counts per minute. What is the half-life of the sample?
Activity ratio 200/800 = (1/2)^n → 1/4 = (1/2)^n → n = 2 half-lives. Since 2 half-lives = 12 hours, half-life = 12 / 2 = 6 hours.
A nucleus of atomic number 90 and mass number 230 emits a beta particle. What is the atomic number of the resulting nucleus?
In beta-minus decay, a neutron converts to a proton, increasing the atomic number by 1. Thus, the new atomic number = 90 + 1 = 91.
A photon of energy 4.0 x 10^-19 J strikes a metal surface with a work function of 2.5 x 10^-19 J. What is the maximum speed of the emitted electron? (m_e = 9.1 x 10^-31 kg)
Maximum kinetic energy KE = E - φ = 4.0 × 10^-19 - 2.5 × 10^-19 = 1.5 × 10^-19 J. KE = (1/2)mv^2 → v = √(2 × KE / m) = √(2 × 1.5 × 10^-19 / 9.1 × 10^-31) ≈ 5.0 × 10^5 ms^-1.
A particle has a de Broglie wavelength of 4 x 10^-10 m when moving at 2 x 10^6 ms^-1. What is the mass of the particle? (h = 6.63 x 10^-34 Js)
De Broglie wavelength λ = h / (mv) → m = h / (λv) = (6.63 × 10^-34) / (4 × 10^-10 × 2 × 10^6) ≈ 8.3 × 10^-31 kg.
A projectile is launched at an angle of 30° with a speed of 20 ms^-1. What is the maximum height reached? (g = 10 ms^-2)
Maximum height H = (u^2 sin^2θ) / (2g) = (20^2 × sin^2 30°) / (2 × 10) = (400 × 0.25) / 20 = 5 m.
A 5 kg mass moves in a circular path of radius 2 m at a constant speed of 4 ms^-1. What is the centripetal force acting on the mass?
Centripetal force F = mv^2 / r = (5 × 4^2) / 2 = (5 × 16) / 2 = 40 N.
A 0.2 kg metal block at 100°C is placed in 0.5 kg of water at 20°C. If the final temperature is 25°C, what is the specific heat capacity of the metal? (Specific heat of water = 4200 Jkg^-1K^-1)
Heat lost by metal = Heat gained by water. m_m c_m (100 - 25) = m_w c_w (25 - 20) → 0.2 × c_m × 75 = 0.5 × 4200 × 5 → c_m = (0.5 × 4200 × 5) / (0.2 × 75) = 800 Jkg^-1K^-1.
Two waves of the same frequency and amplitude interfere destructively. What is the resultant amplitude?
Destructive interference of two waves with the same frequency and amplitude results in complete cancellation, yielding a resultant amplitude of zero.
A convex lens has a focal length of 20 cm. An object is placed 30 cm from the lens. What is the image distance?
Using the lens formula 1/f = 1/u + 1/v, where f = 20 cm, u = 30 cm: 1/20 = 1/30 + 1/v → 1/v = 1/20 - 1/30 = 1/60 → v = 60 cm.
A wire of length 2 m and cross-sectional area 1 x 10^-6 m^2 has a resistance of 4 Ω. What is the resistivity of the material?
Resistivity ρ = RA / L = (4 × 1 × 10^-6) / 2 = 2 × 10^-6 Ωm.
An electron moves at 3 x 10^6 ms^-1 in a magnetic field of 0.1 T, perpendicular to the field. What is the magnetic force on the electron? (q_e = 1.6 x 10^-19 C)
Magnetic force F = qvB = 1.6 × 10^-19 × 3 × 10^6 × 0.1 = 4.8 × 10^-14 N.
A coil rotates in a magnetic field of 0.2 T, producing an emf of 10 V. If the coil has 50 turns and an area of 0.05 m^2, what is the angular speed of the coil?
Emf = NABω → 10 = 50 × 0.05 × 0.2 × ω → ω = 10 / (50 × 0.05 × 0.2) = 40 rad s^-1.
A 5 Ω resistor and a 10 Ω resistor are connected in parallel to a 12 V battery. What is the total current supplied by the battery?
Equivalent resistance R_eq = (5 × 10) / (5 + 10) = 50/15 ≈ 3.33 Ω. Total current I = V / R_eq = 12 / 3.33 ≈ 3.6 A.
A sound wave has a frequency of 500 Hz and a wavelength of 0.68 m. What is the speed of the wave?
Wave speed v = fλ = 500 × 0.68 = 340 ms^-1.
A car of mass 1000 kg accelerates from rest to 20 ms^-1 in 5 s. What is the average power developed by the engine?
Work done = ΔKE = (1/2)mv^2 = (1/2) × 1000 × 20^2 = 200,000 J. Power = Work / time = 200,000 / 5 = 40,000 W = 40 kW.
A concave mirror has a radius of curvature of 40 cm. An object is placed 60 cm from the mirror. What is the image distance?
Focal length f = R/2 = 40/2 = 20 cm. Mirror formula: 1/f = 1/u + 1/v → 1/20 = 1/60 + 1/v → 1/v = 1/20 - 1/60 = 2/60 → v = 30 cm.
A gas at a pressure of 1 x 10^5 Pa and volume of 0.2 m^3 is compressed adiabatically to 0.1 m^3. If the adiabatic index γ = 1.4, what is the final pressure?
For adiabatic process: P_1 V_1^γ = P_2 V_2^γ → P_2 = P_1 (V_1/V_2)^γ = 1 × 10^5 × (0.2/0.1)^1.4 = 1 × 10^5 × 2^1.4 ≈ 2.64 × 10^5 Pa.
An alpha particle is emitted from a nucleus of mass number 238 and atomic number 92. What is the mass number of the resulting nucleus?
An alpha particle has a mass number of 4. Emission reduces the mass number by 4: 238 - 4 = 234.
A 0.3 kg ball is dropped from a height of 8 m. What is its kinetic energy just before hitting the ground? (g = 10 ms^-2)
Potential energy at height = mgh = 0.3 × 10 × 8 = 24 J. At the ground, all potential energy converts to kinetic energy, so KE = 24 J.
A 2 m long string has a mass of 0.01 kg. If it is under a tension of 50 N, what is the speed of a transverse wave on the string?
Wave speed v = √(T/μ), where μ = m/L = 0.01 / 2 = 0.005 kgm^-1. Thus, v = √(50 / 0.005) = √10000 = 100 ms^-1.
A 6 V battery is connected to a 3 Ω resistor. What is the heat generated in the resistor per second?
Power P = V^2 / R = 6^2 / 3 = 36 / 3 = 12 W. Heat generated per second = 12 J.
A light bulb is rated 60 W at 240 V. What is the resistance of the bulb?
Resistance R = V^2 / P = 240^2 / 60 = 57600 / 60 = 960 Ω.
A pendulum of length 0.5 m oscillates on Earth (g = 10 ms^-2). What is its period of oscillation?
Period T = 2π√(L/g) = 2π√(0.5/10) = 2π√0.05 ≈ 2π × 0.224 ≈ 1.4 s.
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