Two forces act on a smooth ring fixed at the origin in the horizontal plane

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

Two forces act on a smooth ring fixed at the origin in the horizontal plane. The first force is

\mathbf{F}_1=40\,\text{N}

directed

30^{\circ}

above the positive

x

-axis. The second force is

\mathbf{F}_2=50\,\text{N}

directed at an angle of

120^{\circ}

from the positive

x

-axis (measured anticlockwise). The ring is to remain in equilibrium under the action of a third force

\mathbf{F}_3.

(i) Express

\mathbf{F}_1

and

\mathbf{F}_2

in terms of their rectangular components. (ii) Using the condition for equilibrium of concurrent forces, obtain the components of the required force

\mathbf{F}_3.

(iii) Hence find the magnitude of

\mathbf{F}_3

and the angle it makes with the positive

x

-axis. (iv) Verify by direct addition of the three vectors that the resultant is indeed zero.