Calculation of Surface Area, Center of Mass, Mass Moment of Inertia and Angular Momentum of Rotating

Question 1: Hyperbolic plate

1.Question 1Consider the thin plate of Figure 1 which is symmetric with respectto they-axis and defined as:{0≤y≤1x+2for0≤x≤1with dimensions expressed in meters. The plate has massm=1kg and is made of ahomogeneous material of densityρ. Determine:- Hyperbolic platea) the surface area of the plate and the material density b) the position of the plate centre of massc) the plate mass moment of inertia with respect to thexaxis,d) the plate mass moment of inertia with respect to theyaxis,e) the plate mass moment of inertia with respect to the axis perpendicularto the plate and passing through the plate centre of mass.Version 1Page2of5 2.Question 2 Consider the thin homogeneous square plate of massm=1kgand of sidea=0.50m shown in Figure 2. The plate rotates at constant angular velocity,ω=100rad/s, about an axis parallel to one of its side and at a distancea/4, as shown.The bearingsAandBare friction-less and located at distancel/2(l>a) from the platevertical centreline, as shown in Figure 2.xyzare orthogonal axes attached to the platewith unit vectorsi,jandk.Figure 2: Rotating rectangular platea) Write down the components of the plate angular velocity,ω, in the body axis systemb) Calculate the angular momentum of the plate with respect toAc) Deduce from b) the expression of the time rate of change of the angular momentum.]d) Find the magnitude of the dynamic reaction forces at the bearings