A boy is dragging a bag of mass $8 \mathrm{kg}$ up a slope at an angle of $17°$ to the horizontal and exerts a force of $50 \mathrm{N}$ parallel to the slope to do this. Air resistance, $F,$ parallel to the slope prevents the boy from increasing his speed, so he maintains a constant speed. Find the magnitude of the air resistance and the normal contact force. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A heavy box of mass $50 \mathrm{kg}$ is on a slope at angle $25°$ to the horizontal. There is no friction to prevent it from sliding down the slope, but there are three rods attached, at $40°, 50°$ and $60°$ above the slope, for people to drag it. A man and two boys hold the rods to keep the box in equilibrium.

Show that, if the man pulls with a force of $170 \mathrm{N}$ and each boy can pull with a force of up to $90 \mathrm{N},$ they can hold the box in equilibrium. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A heavy box of mass $50 \mathrm{kg}$ is on a slope at angle $25°$ to the horizontal. There is no friction to prevent it sliding down the slope, but there are three rods attached, at $40°, 50°$ and $60°$ above the slope, for people to drag it. A man and two boys hold the rods to keep the box in equilibrium.

If the man pulls with force $180 \mathrm{N}$ and each boy can pull with a force up to $70 \mathrm{N},$ determine whether or not they can hold the box in equilibrium and state which rod each should hold. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A box of mass $20 \mathrm{kg}$ is on a horizontal surface. There are three rods attached on one side, at $10°, 25°$ and $35°$ above the horizontal, for people to drag it. Three people are available to pull on these rods and they are capable of providing forces of $150 \mathrm{N}, 200 \mathrm{N}$ and $250 \mathrm{N}.$

The box is being pulled in the opposite direction by a horizontal force of $425 \mathrm{N}.$ Show that only two of the people are required to keep the box in equilibrium. State which of the rods each person holds.

A box of mass $20 \mathrm{kg}$ is on a horizontal surface. There are three rods attached on one side, at $10°, 25°$ and $35°$ above the horizontal, for people to drag it. Three people are available to pull on these rods and they are capable of providing forces of $150 \mathrm{N}, 200 \mathrm{N}$ and $250 \mathrm{N}.$

The horizontal force is increased to $550 \mathrm{N}$. Show that if the box is to be prevented from moving horizontally, it cannot remain on the ground.

A particle is held in place by forces of $8\mathrm{N},11\mathrm{N}$ and $12\mathrm{N},$ as shown in the diagram. Find the values of $\mathrm{\theta }$ and $\mathrm{\phi }.$

A mass of $5 \mathrm{kg}$ is held in equilibrium by two ropes with tensions of $30 \mathrm{N}$ and $40 \mathrm{N}.$ Find the angles that the ropes make with the vertical.