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
I., let's knock this out. A solid disk of mass M = 5.0 \, kg and radius R = 0.20 \, m is initially at rest. A constant torque = 10.0 \, N · m is applied to the disk about its central axis. a) What is the angular acceleration of the disk? b) What is the angular velocity of the disk after 5.0 \, s? c) How many revolutions does the disk make in 5.0 \, s? Step 1: Calculate the moment of inertia for a solid disk. The formula for the moment of inertia of a solid disk is I = (1)/(2)MR^2. I = (1)/(2)(5.0 \, kg)(0.20 \, m)^2 I = (1)/(2)(5.0 \, kg)(0.040 \, m^2) I = 0.10 \, kg · m^2 Step 2: Calculate the angular acceleration using Newton's second law for rotation. The formula is = I. 10.0 \, N · m = (0.10 \, kg · m^2) = 10.0 \, N · m0.10 \, kg · m^2 = 100 \, rad/s^2 Step 3: Calculate the angular velocity after 5.0 \, s. The formula for final angular velocity is _f = _i + t. Since the disk starts from rest, _i = 0. _f = 0 + (100 \, rad/s^2)(5.0 \, s) _f = 500 \, rad/s Step 4: Calculate the angular displacement after 5.0 \, s. The formula for angular displacement is = _i t + (1)/(2) t^2. = (0)(5.0 \, s) + (1)/(2)(100 \, rad/s^2)(5.0 \, s)^2 = 0 + (1)/(2)(100 \, rad/s^2)(25.0 \, s^2) = 1250 \, rad Step 5: Convert the angular displacement from radians to revolutions. There are 2 radians in one revolution. Revolutions = 1250 \, rad2 \, rad/rev Revolutions ≈ 198.9 \, rev a) The angular acceleration is 100 \, rad/s^2. b) The angular velocity after 5.0 \, s is 500 \, rad/s. c) The number of revolutions in 5.0 \, s is approximately 198.9 \, rev. Got more? Send 'em 📸