f[
x_
]:=
x Sin[1/x]

D[f[x],x]

Solve[0==D[f[x],x],x]

Plot[f[x],{x,-0.1,0.1}
,PlotRange->Full
]

Galileo’s rolling ball

A ball of radius R rolling at velocity has angular velocity ω=(v/R) (in radians/s). The kinetic energy of rotation is (1/2)I (ω^2). For a solid sphere, I=(2/5)m (R^2). So the total kinetic energy of a rolling ball is

te1=(1/2)m (v^2)

re1=(1/2)((2/5)m (r^2))((((v/r)))^2)

((te1+re1)/te1)

g=.

Solve[m g h==te1+re1, v]

Solve[m g h==te1, v]

Floor and fractional part

Block[{prad=0.07},
Plot[Floor[x],{x,-2,3}
,AspectRatio->Automatic
,Epilog->{
Table[
Disk[{x,Floor[x]},prad],
{x,-3,2}
],
Table[
Circle[{x+1,Floor[x]},prad],
{x,-3,2}
]
}
]
]

Block[{prad=0.07},
Plot[x-Floor[x],{x,-2,3}
,AspectRatio->Automatic
,Epilog->{
Table[
Disk[{x,0},prad],
{x,-2,2}
],
Table[
{
White,Disk[{x+1,1},prad],
Black,Circle[{x+1,1},prad]
},
{x,-2,2}
]
}
]
]