An airplane sitting on a grassy field has a single propeller consisting of 3 identical blades, as shown in the diagram. Each blade has mass M, = 20 kg and length L = 0.8 m. The mass of each blade is not evenly distributed, so that the center of mass is located e= 0.3 m from the outer end of the blade. The moment of inertia of each blade about its center of mass is Iem =2 kg m². a) The propeller is initially at rest. The pilot then starts the engine and after 5 s the propeller is rotating at o = 30 radians/s. What is the magnitude of the angular acceleration a during that time assuming the angular acceleration is constant? blade b) What is the moment of inertia of the entire propeller about its axis of rotation? C.m. c) What is the magnitude of the net torque on the propeller during the initial 5 second period? blade d) At t = 5 s, when the propeller is spinning with angular velocity o = 30 radians/s, what is the speed of the outer tip of each blade of the propeller? C.m. -C.m. blade

b) Find the moment of inertia tensor of a solid hemisphere of mass M and radius R centered at the origin, where the base of the hemisphere is placed on the z = 0 plane and is facing towards the -z- axis and. x Z (0,0,0) ***** *********** R L ********************* y

1. A circular disk has a density p(r) which varies as a function of distance from the center of the disk. The disk has a radius R and a thickness t. Assuming p(r) = Po(1-r/R): Calculate the total mass of the disk. a. b. Calculate the moment of inertia of the disk about an axis out of the plane of the page and passing through Point G. C. Calculate the moment of inertia of the disk about an axis out of the plane of the page and passing through Point O. G R

6. Derive an expression for the moment of inertia of the pictured semi- circular loop-rod device about an axis through the point P, perpen- dicular to the page. The mass of the semicircular loop is 3.14 kg, and the rod is 3.0 kg. Take R = 1.0 m. [24.14 kg m²] 3R

Find the equivalent moment of inertia of the rocker arm assembly with respect to the location of kt mass1=5kg Jo=20kg-m2 mass2=2kg, a is half of b. The rocker is 3 meters long. (kg-m2) *

An oversized yo-yo is made from two identical solid disks each of mass M - 1.60 kg and radius R- 9.0 cm. The two disks are joined by a solid cylinder of radius r- 4.00 cm and mass m - 1.00 kg as in the figure below. Take the center of the cylinder as the axis of the system. (a) What is the moment of inertia of the system? Give a symbolic answer. (Use any variable or symbol stated above as necessary.) moment of inertia- MR + (b) What torque does gravity exert on the system with respect to the given axis? V N-m Take downward as the negative coordinate direction. (c) As depicted in the figure above, is the torque exerted by the tension positive or negative? (View the yo-yo from the left and let the countercdockwise direction be positive.) positive O negative (d) Is the angular acceleration positive or negative? O positive O negative (e) What about the translational acceleration? O positive O negative () Write an equation for the angular acceleration a (alpha) in terms of the translational…

Consider a wheel of radius r = 0.28 m, with six spokes (or three rods of length equal to the diameter of the wheel). The total mass of the wheel is 3.0 kg and 55% of the mass is evenly distributed on the circumference while the rest is evenly distributed on the spokes. Calculate the moment of inertia. r

Examples: Moments of inertia of the pendulum about an axis 1) passing through point O, and 2) mass center G of the pendulum. Both rods OA and BC are uniform. Rod OA weighs 10 lb, and BC weighs 8 lb. B A y 2 ft 1 ft1 ft C

5. Consider the disk with a rectangular cutout shown in the figure below. The disk has the dimensions shown and has a thickness of 3 m and a density of 4000 kg/m³. a. Calculate the total mass of the disk. b. Calculate the x and y coordinates of the disk's center of gravity from Point O. C. Calculate the disk's moment of inertia about the z-axis passing through Point O. thickness of 3 m y 10 m ELDI 3 m 6 m

Match the most appropriate form of the equation for the moment of inertia to the image shown. All objects are rotating about point O. Note: ris distance, m is a mass, k is a radius of gyration and d is a distance from the mass moment of inertia IG about the center of mass A. D. O M X Mass element Z Rotation axis X A.1=1+md² G B. = SM √² r²dm C.I=mk²_ D.1= Σm,r?