A box with weight $400 \mathrm{N}$ is at rest on a horizontal surface. A man is pulling on a rope to try to get the box to move. The force he can exert depends on the angle at which he holds the rope, so that when the rope is at an angle $1600\sin \theta \mathrm{N}$ above the horizontal, the force he exerts is $1600\sin \theta \mathrm{N}.$ He starts by holding the rope horizontally and gradually increases the angle, thereby increasing the force. Another man tries to prevent this motion of the box, by pulling horizontally. He can exert a maximum force of $700 \mathrm{N}.$ Find the angle at which the box can no longer remain on the ground. Hence, determine whether the box lifts off the ground first or slides along the ground first.

A particle has three forces acting on it, as shown in the diagram, where $\sin \theta =\frac{3}{5}.$ Show that $\sin \theta =\frac{3}{5}$ by resolving horizontally, and write down another equation by resolving vertically. Hence, show that $G=75\sqrt{3}+100$ and find $F.$

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A particle has three horizontal forces acting on it, as shown in the diagram. Show that $\cos \alpha =\frac{14-13\cos \beta }{15}$ and find an expression for $\sin \alpha$. Use ${\cos }^2\alpha +{\sin }^2\alpha =1$ to get an equation in $\beta$. Hence, find $\alpha$ and $\beta$.

Find the component of the following force in the direction of the dashed arrow. Although it might seem clear from the diagram, make sure you specify whether the component is in the given direction or in the opposite direction.

Find the component of the following force in the direction of the dashed arrow. Although it might seem clear from the diagram, make sure you specify whether the component is in the given direction or in the opposite direction.

Find the components of the following forces in the direction of the dashed arrow. Although it might seem clear from the diagram, make sure you specify whether the component is in the given direction or in the opposite direction.

Find the components of the following forces in the direction of the dashed arrow. Although it might seem clear from the diagram, make sure you specify whether the component is in the given direction or in the opposite direction.

Find the component of the following force perpendicular to the direction of the dashed arrow. Make it clear whether the component is in the perpendicular direction clockwise or anticlockwise from the direction given.

Find the component of the force perpendicular to the direction of the dashed arrow. Make it clear whether the component is in the perpendicular direction clockwise or anticlockwise from the direction given.