Force — Question 5

Question 5

Force components depend on axes; the force does not

A measured force has ground components $(4,3)$ N. New axes are rotated $90^\circ$ counterclockwise, so $\hat e_{x^{\prime}}=\hat j$ and $\hat e_{y^{\prime}}=-\hat i$.

1. Find the components in the new axes.

2. Check the magnitude in both descriptions.

3. Would summing old x and new y components form a legitimate component pair?

Question 5 - Solution

Force components depend on axes; the force does not

(a) Project on the new basis.

$$F_{x'}=\vec F\cdot\hat j=3,\quad F_{y'}=\vec F\cdot(-\hat i)=-4\,\mathrm N$$ $F_{x^{\prime}}=3$ N and $F_{y^{\prime}}=-4$ N, so the new pair is $(3,-4)$ N.

(b) Test the invariant magnitude.

$$\boxed{|\vec F|=\sqrt{4^2+3^2}=\sqrt{3^2+(-4)^2}=5\,\mathrm N}$$ Both $\sqrt{4^2+3^2}$ and $\sqrt{3^2+(-4)^2}$ equal 5 N. Only the description has changed.

(c) Use a single basis.

$$\vec F=F_x\hat i+F_y\hat j=F_{x'}\hat e_{x'}+F_{y'}\hat e_{y'}$$ No. The proposed components belong to different axes and are not the coordinates of the same vector in one orthonormal basis. Transform the full vector first, then carry out any component arithmetic.