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.
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a) Given: Mass m = 20 kg Vertical height h = 0.9 m Acceleration due to gravity g = 9.8 m/s^2 i) Determine the potential energy at its position. Step 1: The formula for potential energy is PE = mgh. Step 2: Substitute the given values into the formula. PE = (20 kg) × (9.8 m/s^2) × (0.9 m) Step 3: Calculate the potential energy. PE = 196 N × 0.9 m PE = 176.4 J The potential energy at its position is 176.4 J. ii) Determine the speed of the body when passing through the lowest point. Step 1: Apply the principle of conservation of mechanical energy. At the initial position, all energy is potential energy (PE_i = mgh, KE_i = 0). At the lowest point, all potential energy is converted to kinetic energy (PE_f = 0, KE_f = (1)/(2)mv^2). PE_i + KE_i = PE_f + KE_f mgh + 0 = 0 + (1)/(2)mv^2 Step 2: Simplify the equation and solve for v. mgh = (1)/(2)mv^2 gh = (1)/(2)v^2 v^2 = 2gh v = sqrt(2gh) Step 3: Substitute the values and calculate the speed. v = sqrt(2 × 9.8 m/s)^2 × 0.9 m v = sqrt(17.64 m)^2/s^2 v = 4.2 m/s The speed of the body when passing through the lowest point is 4.2 m/s. b) Given: Load mass m = 2000 kg Vertical distance d = 3.0 m Time t = 6 s Rated power of electric motor P_in = 12.5 kW Acceleration due to gravity g = 9.8 m/s^2 i) Determine the work done by the crane. Step 1: The formula for work done against gravity is W = Fd = mgh. Step 2: Substitute the given values. W = (2000 kg) × (9.8 m/s^2) × (3.0 m) Step 3: Calculate the work done. W = 19600 N × 3.0 m W = 58800 J The work done by the crane is 58800 J. ii) Determine the power developed by the crane. Step 1: The formula for power is P = (W)/(t). Step 2: Substitute the work done from part (b)(i) and the given time. P = 58800 J6 s Step 3: Calculate the power. P = 9800 W P = 9.8 kW The power developed by the crane is 9.8 kW. iii) Determine the efficiency of the crane. Step 1: The formula for efficiency is = P_outP_in × 100\%. Step 2: Convert the input power to Watts: P_in = 12.5 kW = 12.5 × 1000 W = 12500 W. The output power P_out is the power developed by the crane, which is 9800 W from part (b)(ii). Step 3: Substitute the output power and input power into the efficiency formula. = 9800 W12500 W × 100\% Step 4: Calculate the efficiency. = 0.784 × 100\% = 78.4\% The efficiency of the crane is 78.4\%. Send me the next one 📸