The Vector Nature of Forces

A problem statement says: "Ana pushes the trolley with a force of $50\ \text{N}$ to the east." State the magnitude, direction, and sense of this force, and explain why we say that force is a vector quantity and giving only its magnitude is not enough.

Three forces act on a box along the same line: $12\ \text{N}$ to the right, $7\ \text{N}$ to the right and $5\ \text{N}$ to the left. Calculate the resultant force and state its direction.

Two workers push a container in the same direction and sense, one with a force of $300\ \text{N}$ and the other with a force of $250\ \text{N}$. Calculate the resultant force on the container and represent the situation with a diagram in your notebook.

In a game of tug-of-war, team A pulls to the left with a force of $450\ \text{N}$ and team B pulls to the right with $400\ \text{N}$, as shown in the figure.

Calculate the magnitude of the resultant force and state towards which team the rope will move.

A horse pulls a cart always with the same force of $600\ \text{N}$, as shown in the figure.

Explain whether the force that moves the cart forward is the same when the horse pulls in the direction of the path as when it pulls at an angle to it. What data would you need to calculate the effective force in the second case?

The components of the force exerted by a drone's motor on the air are $F_x = 12\ \text{N}$ and $F_y = 5\ \text{N}$, as shown in the figure.

1. a) Calculate the magnitude of the resultant force.
2. b) Calculate the angle it forms with the horizontal using $\arcsen$.

A robot vacuum cleaner collides with a corner and receives two perpendicular pushes simultaneously, such that the components of the total force it receives are $(9,\ 12)\ \text{N}$. Calculate the magnitude of the resultant force on the robot.

Three concurrent forces act on a piece of furniture: $10\ \text{N}$ to the east, $6\ \text{N}$ to the west and $3\ \text{N}$ to the north, as shown in the figure.

1. a) First, calculate the resultant of the two horizontal forces.
2. b) Combine it with the vertical force and find the magnitude of the overall resultant.
3. c) Calculate the angle formed by the resultant with the horizontal using $\arcsen$.

A boat is pushed by its engine with $20\ \text{N}$ to the east and by the wind with $10\ \text{N}$ forming $60^\circ$ with the east, as shown in the figure.

Using $\sen 60^\circ \approx 0.87$ and $\cos 60^\circ = 0.5$:

1. a) Resolve the wind force into east and north directions.
2. b) Add the components of both forces along each axis.
3. c) Calculate the magnitude of the resultant force on the boat (give the answer to one decimal place).

A local council is studying two designs for an accessible ramp. A wheelchair with its user weighs $800\ \text{N}$ in total, as shown in the figure.

1. a) Calculate the component of the weight parallel to the ramp if it forms $15^\circ$ with the horizontal ($\sen 15^\circ \approx 0.26$).
2. b) Calculate the same component for a $30^\circ$ ramp ($\sen 30^\circ = 0.5$).
3. c) In light of the results, explain which design is suitable for an accessible ramp and why.

The anchor point of a zip wire is in equilibrium under the action of three forces. Two of them are $8\ \text{N}$ horizontally and $6\ \text{N}$ vertically.

1. a) Calculate the magnitude of the resultant of these two forces.
2. b) Deduce the magnitude and direction that the third force must have so that the sum of the three is zero. Justify your answer.

Two tugboats pull a ship, each with a force of $500\ \text{N}$, using cables that form $30^\circ$ with the ship's direction of motion, one on each side, as shown in the figure.

Use $\cos 30^\circ \approx 0.87$ and $\sin 30^\circ = 0.5$.

1. a) Calculate the component of each tugboat's force in the direction of motion.
2. b) Explain what happens to the sideways components of both forces.
3. c) Calculate the total force moving the ship forward.