Problem 3

Clear["Global`*"]

eqs={
n== m1 g Cos[θ] ,
m1 a == m1 g Sin[θ] - tension,
tension == (1/2)m2 a};

Solve[eqs,{n,a,tension}]

Problem 5

Clear["Global`*"]

PlanetData[]

values={
massEarth->PlanetData["Earth","Mass"],
radiusEarth->UnitConvert[PlanetData["Earth","Radius"],"SI"],
gSurface->UnitConvert[PlanetData["Earth","Gravity"],"SI"],
newtonG->Quantity[ "G"]
}

eq1 = m1 g == ((newtonG massEarth m1 )/(r^2));
solg= Solve[eq1,g][[1]]

grav[r_]=g  /.solg/.values

Compute from surface of Earth

radiusEarth /.values

earthG=grav[radiusEarth] /.values //UnitConvert

Compute from surface of Earth + 100 Miles

grav[radiusEarth+Quantity[100 ,"Miles"]] /.values //UnitConvert

%/earthG

Compute from moon

moonRadius= PlanetaryMoonData["Moon","AverageOrbitDistance"]

radius = PlanetaryMoonData["Moon","AverageOrbitDistance"];
grav[radius/.values ]//UnitConvert

%/earthG

Assume g=9.8, compute time to fall from moon:

solT=Solve[ x == (1/2) a (t^2),t][[2]]

time1= t /.solT /.{x ->PlanetaryMoonData["Moon","AverageOrbitDistance"], a->earthG}

UnitConvert[time1,"Hours"]

UnitConvert[time1,"Days"]

Assume g=((massEarth newtonG)/(r^2)), compute time to fall from moon:

eq1= ((- newtonG massEarth mass )/r0)== ((- newtonG massEarth mass )/(r[t]))+ (1/2) mass ((r'[t])^2)

sol=Solve[ eq1, r'[t]][[1]] //Simplify

eq2= (dr/dt) ==( r'[t]/.sol) /.{r[t]->r}

solt=Solve[eq2,dt][[1]]

integrand = dt /.solt /.{dr->1}

time= Integrate[ integrand,{r,r0,0},Assumptions->{Re[r0]>0&&Im[r0]==0&& massEarth newtonG>0}] //PowerExpand

time=time  /.values /.r0->moonRadius //UnitConvert

UnitConvert[time,"Hours"]

UnitConvert[time,"Days"]