Exercises


  1. Calculate the horizontal and vertical components of the following forces:
    1. A force of \( 150 \) Newtons exerted at an angle of \( 75^{\circ} \) to the horizontal.
    2. A force of \( 150 \) Newtons exerted at an angle of \( 15^{\circ} \) to the horizontal.
    3. A force of \( 20 \) Newtons exerted at an angle of \( 28^{\circ} \) to the horizontal.
    4. A force of \( 300 \) Newtons exerted at an angle of \( 45^{\circ} \) to the horizontal.
     
  2. Find the magnitude and direction of the resultant of the following forces:
    1. \( 20 ~\text{N}~ [N]\) and \( 25 ~\text{N}~ [E]\)
    2. \( 12 ~\text{N}~ [S]\) and \( 10 ~\text{N}~ [SE]\)
    3. \( 50 ~\text{N}~ [N]\), \( 90 ~\text{N}~ [S]\) and \( 35 ~\text{N}~ [E]\)
  3. Which of the following sets of forces could act on an object and produce equilibrium?
    1. \( \left \lvert \vec{u} \right \rvert = 12, \left \lvert \vec{v} \right \rvert = 8, \left \lvert \vec{w} \right \rvert = 15 \)
    2. \( \left \lvert \vec{u} \right \rvert = 10, \left \lvert \vec{v} \right \rvert = 6, \left \lvert \vec{w} \right \rvert = 20 \)
    3. \( \left \lvert \vec{u} \right \rvert = 34, \left \lvert \vec{v} \right \rvert = 12, \left \lvert \vec{w} \right \rvert = 46 \)
  4. Find the angle between \( \vec{a} \) and \( \vec{b} \) if \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \) and the magnitudes are as follows:
    1. \( \left \lvert \vec{a} \right \rvert = 12 ~\text{N}~, \left \lvert \vec{b} \right \rvert = 5 ~\text{N}~, \left \lvert \vec{c} \right \rvert = 13 ~\text{N}~ \)
    2. \( \left \lvert \vec{a} \right \rvert = 10 ~\text{N}~, \left \lvert \vec{b} \right \rvert = 20 ~\text{N}~, \left \lvert \vec{c} \right \rvert = 22 ~\text{N}~ \)
    3. \( \left \lvert \vec{a} \right \rvert = 32 ~\text{N}~, \left \lvert \vec{b} \right \rvert = 40 ~\text{N}~, \left \lvert \vec{c} \right \rvert = 65 ~\text{N}~ \)
  5. Three forces, \( \vec{u}, \vec{v}, \vec{w} \), act on an object to produce equilibrium. Calculate the magnitude and direction of \( \vec{w} \), given the following vectors \( \vec{u} \) and \( \vec{v} \):
    1. \( \vec{u} = 25 ~\text{N}~ [N], \vec{v} = 30 ~\text{N}~ [E] \)
    2. \( \vec{u} = 100 ~\text{N}~ [N], \vec{v} = 50 ~\text{N}~ [E] \)
    3. \( \vec{u} = 65 ~\text{N}~ [SE], \vec{v} = 90 ~\text{N}~ [N] \)
  6. A sailboat experiences a force of \( 100~\text{N}\) in the direction \( [NW] \) from the wind, while a tidal current is exerting a force of \( 45 ~ \text{N} \) in a southerly direction.
    1. What is the magnitude of the resultant force on the sailboat?
    2. In which direction will the sailboat move?
  7. A car weighing \( 2500~\text{kg} \) is stopped on a road inclined at \( 12^{\circ} \) to the horizontal.
    1. What is the magnitude of the force which prevents it from sliding down the hill?
    2. What force is being exerted perpendicular to the road surface?
  8. An object weighing \( 20~\text{kg} \) is suspended by two wires of equal length \( 50~\text{cm} \). How far apart must they be attached to the surface above so that the force on each is \( 150~\text{N} \)?
  9. An outdoor chair is suspended from a tree by a single cable of length \( 1.5~\text{m} \). The mass of the chair and the person sitting in it is a total of \( 85~\text{kg} \). If someone pulls the chair through a horizontal distance of \( 80~\text{cm} \) and holds it there:
    1. What is the increase in the force exerted by the cable?
    2. How much sideways force must be exerted to hold the chair at that location?
  10. A sled is pulled forward by two dogs, each attached to the sled by a tow line. The angle between the two tow lines is \( 35^\circ \). How many times harder must one dog be pulling if the sled's path is at an angle of \( 20^\circ \) to the left hand dog's towline?
  11. A balloon is tethered to the ground by a rope. The balloon is experiencing an upwards force of \( 10~\text{N} \) because of its buoyancy, and a horizontal force of \( 15~\text{N} \) because of an incoming breeze.
    1. How much force must the tether exert in order to keep the balloon still?
    2. What angle must the tether make with the ground?
  12. An inflatable sign is anchored to the ground by a steel cable. The sign has a \( 10~\text{N} \) force acting on it in the \( [up] \) direction, a \( 12~\text{N} \) force acting on it in the \( [N] \) direction, and a \( 15~\text{N} \) force acting on it in the \( [W] \) direction.
    1. Find the magnitude of the resultant.
    2. Find the angle which the resultant makes with each of the three component forces.