Forces on an oscillating skateboard

When you do this sort of analysis it's important you choose a clear sign convention then adhere to it strictly.

In this case the radial coordinate $r$ is the distance from the centre of the circle. $r=0$ at the centre and it increases as we move outwards away from the centre, so for the radial component outwards away from the centre is positive and inwards towards the centre is negative. If you look at your diagram you'll see that for $-pi/2 < varphi < pi/2$ the gravitational force on the skateboard points outwards away from the centre so it is positive i.e. $+mgcosvarphi$. Likewise the normal force on the skateboard points inwards towards the centre so it is negative i.e. $-N$.

If you manage to skate so fast you rise higher than the centre of the circle the gravitational force now points inwards and becomes negative, but you don't have to worry about this as the sign of $cosvarphi$ becomes negative for $varphi < -pi/2$ and $varphi > pi/2$ so your equation automatically takes care of this for you.

For the angular coordinate the convention is that anticlockwise is positive (why this should be is lost in the mists of time). You are taking $varphi=0$ for the vertically downward direction, and moving anticlockwise is the positive direction, so any tangential force is positive if it points anticlockwise and negative if it points clockwise. When you are to the right of the centre, i.e. $0 < varphi < pi/2$, the tangential force points left and that is the clockwise direction. Hence the tangential force is $-mgsinvarphi$.

When you move the the left of the bottommost point the tangential force points right/anticlockwise so it is positive, but again this is taken care of automatically because the sign of $sinvarphi$ changes to negative for $-pi/2 < varphi < 0$.

With experience most of us become casual about signs because our intuition tells us what is correct, but when you are first learning mechanics it makes life a lot easier if you are careful to define your sign convention and stick to it.

Put coordinate system at the center of the half circle

the position vector to the mass is

$$vec R=r,left[ begin {array}{c} sin left( varphi right) -cos left( varphi right) end {array} right] $$

thus:

$$vec e_r=left[ begin {array}{c} sin left( varphi right) -cos left( varphi right) end {array} right] vec e_varphi= left[ begin {array}{c} cos left( varphi right) sin left( varphi right) end {array} right] vec e_rcdot vec e_varphi=0quad,vec e_rcdot vec e_r=1quad,vec e_varphicdot vec e_varphi=1 $$

you want to "write" the the force like this

$$ vec F=F_r,vec e_r+F_varphi,vec e_varphi$$

with: $$vec F=N,begin{bmatrix} -sin(varphi) cos(varphi) end{bmatrix}+ m,gbegin{bmatrix} 0 -1 end{bmatrix}$$

from here

$$F_r=vec Fcdot vec e_r=m,gcos(varphi)-N F_varphi=vec Fcdot vec e_varphi=-m,g,sin(varphi)$$