This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Hey hafsatjibril2006, good to see you again. To find the maximum distance the spring will be compressed, we use the principle of conservation of energy. The initial gravitational potential energy of the block is converted into elastic potential energy in the spring and gravitational potential energy relative to the lowest point of compression. Step 1: Define the initial and final states and the energies involved. Let m be the mass of the block, h be the initial height above the spring, K be the spring constant, g be the acceleration due to gravity (9.8 m/s^2), and x be the maximum compression of the spring. Initial state (block at height h above the uncompressed spring): The block has gravitational potential energy. We set the reference point for potential energy at the maximum compression of the spring. So, the total initial height of the block relative to this lowest point is h+x. Initial potential energy (PE_initial) = mg(h+x) Initial kinetic energy (KE_initial) = 0 (since it's dropped) Total initial energy (E_initial) = mg(h+x) Final state (spring maximally compressed by x): The block is momentarily at rest, so KE_final = 0. The spring is compressed by x, so it has elastic potential energy (PE_elastic) = (1)/(2)Kx^2. At this point, the block is at the reference height for gravitational potential energy, so PE_gravitational, final = 0. Total final energy (E_final) = (1)/(2)Kx^2 Step 2: Apply the conservation of energy principle. E_initial = E_final mg(h+x) = (1)/(2)Kx^2 Step 3: Substitute the given values into the equation. Given: m = 4 kg h = 0.80 m K = 1916 N/m g = 9.8 m/s^2 4 kg × 9.8 m/s^2 × (0.80 m + x) = (1)/(2) × 1916 N/m × x^2 39.2 (0.80 + x) = 958 x^2 31.36 + 39.2x = 958x^2 Step 4: Rearrange the equation into a standard quadratic form (ax^2 + bx + c = 0). 958x^2 - 39.2x - 31.36 = 0 Step 5: Solve the quadratic equation for x using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a = 958, b = -39.2, and c = -31.36. x = -(-39.2) ± sqrt((-39.2)^2 - 4(958)(-31.36))2(958) x = 39.2 ± sqrt(1536.64 + 120000.64)1916 x = 39.2 ± sqrt(121537.28)1916 x = (39.2 ± 348.622)/(1916) We get two possible values for x: x_1 = (39.2 + 348.622)/(1916) = (387.822)/(1916) ≈ 0.2024 m x_2 = (39.2 - 348.622)/(1916) = (-309.422)/(1916) ≈ -0.1615 m Step 6: Choose the physically meaningful solution. Since x represents a distance of compression, it must be a positive value. Therefore, x ≈ 0.2024 m. Rounding to three significant figures: x ≈ 0.202 m The maximum distance through which the spring will be compressed is: 0.202 m What's next?