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 Jordan, good to see you again. Let's solve Question 7, taking into account the dimensions provided in Figure 7. QUESTION 7: (10 marks) A full steel cylinder of 2000 \, mm diameter is turning about a fixed axis (GZ) with the rotational velocity of 1500 \, rev/min. The volume density of the steel is = 7800 \, kg/m^3. Recall: J_GZ = (MR^2)/(2). From Figure 7: Diameter D = 2000 \, mm = 2.000 \, m Length L = 500 \, mm = 0.500 \, m a) Determine the volume V of the cylinder. Step 1: Calculate the radius and then the volume of the cylinder. R = (D)/(2) = 2.000 \, m2 = 1.000 \, m V = R^2 L V = (1.000 \, m)^2 (0.500 \, m) V = 0.5 \, m^3 V ≈ 1.571 \, m^3 b) Calculate the mass M of the cylinder. Step 2: Calculate the mass of the cylinder using its volume and the given density. M = V M = (7800 \, kg/m^3)(0.5 \, m^3) M = 3900 \, kg M ≈ 12252 \, kg c) Determine the moment of inertia J_GZ of the cylinder. Step 3: Calculate the moment of inertia using the given formula, mass, and radius. J_GZ = (MR^2)/(2) J_GZ = (3900 \, kg)(1.000 \, m)^22 J_GZ = 1950 \, kg · m^2 J_GZ ≈ 6126 \, kg · m^2 d) Express the angular velocity . Step 4: Convert the rotational velocity from revolutions per minute to radians per second. = 1500 \, rev/min × 2 \, rad1 \, rev × 1 \, min60 \, s = (1500 × 2)/(60) \, rad/s = 50 \, rad/s ≈ 157.1 \, rad/s e) Determine the kinetic energy K_E. Step 5: Calculate the rotational kinetic energy using the moment of inertia and angular velocity. K_E = (1)/(2) J_GZ ^2 K_E = (1)/(2) (1950 \, kg · m^2) (50 \, rad/s)^2 K_E = (1)/(2) (1950) (2500^2) \, J