Solve the following equation numerically using the forward Euler method du =t²-10 with the initial condition u(0)=50 dt As part of the solution process converge the solution by decreasing the step size You can determine the solution by taking the root-mean-square of the error with respect to the exact solution. You will need to calculate the solution to this differential equation. Then for each step size h, you can calculate the root-mean-square of the error. Once the root-mean-square of the error is less than the accuracy of the solution, your result is converged. Choose a time grid say from 0 to 20. Once you have this working, try solving using the Forward Euler method, du dt =u*cos(t) with the initial condition u(0)=1, and using a mesh from [0,10].

(Conversion) An object’s polar moment of inertia, J, represents its resistance to twisting. For a cylinder, this moment of inertia is given by this formula: J=mr2/2+m( l 2 +3r 2 )/12misthecylindersmass( kg).listhecylinderslength(m).risthecylindersradius(m). Using this formula, determine the units for the cylinder’s polar moment of inertia.

(Conversion) Determine which of the following equations can’t be valid because they yield incorrect unit measurements: a.F=mab.F=m( v 2 /t)c.d=( at 2 )d.d=vte.F=mvtF=force(N)m=mass( kg)a=acceleration( m/s 2 )v=velocity( m/s)t=time(s)

(Thermodynamics) The work, W, performed by a single piston in an engine can be determined by this formula: W=Fd F is the force provided by the piston in Newtons. d is the distance the piston moves in meters. a. Determine the units of W by calculating the units resulting from the right side of the formula. Check that your answer corresponds to the units for work listed in Table 1.1. b. Determine the work performed by a piston that provides a force of 1000 N over a distance of 15 centimeters.