Vector Nature of Forces: Graphical and Numerical Operations

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.

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?

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.

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 perpendicular forces of $3\ \text{N}$ and $4\ \text{N}$ act on the same point of a toy. Draw the situation in your notebook and calculate the magnitude of the resultant force using Pythagoras' theorem.

You can do it on paper and send a photo with your answer.

The wind pushes a boat with $6\ \text{N}$ to the east, and the current pushes it with $8\ \text{N}$ to the north, as shown in the figure.

Calculate the magnitude of the resultant force on the boat.

Two forces of $5\ \text{N}$ each act on the same point forming an angle of $90^\circ$. Draw the situation in your notebook using a scale of $1\ \text{cm}$ per newton, apply the parallelogram rule and measure the magnitude of the resultant. Then check the result using Pythagoras' theorem (it should be approximately $7.1\ \text{N}$).

You can do it on paper and send a photo with your answer.

A ski lift pulls a skier with a force of $200\ \text{N}$ that forms $30^\circ$ with the horizontal, as shown in the figure.

Calculate the horizontal and vertical components of the force (use $\sen 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.87$) and state which of the two moves the skier along the slope.

The string of a bow propels the arrow with a force of $20\ \text{N}$ that forms $60^\circ$ with the horizontal. Calculate the horizontal and vertical components of that force (use $\sen 60^\circ \approx 0.87$ and $\cos 60^\circ = 0.5$).

A child pulls a sledge with a rope forming $37^\circ$ with the ground, exerting a force of $50\ \text{N}$, as shown in the figure.

Using $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$:

1. a) Calculate the component of the force that moves the sledge forward.
2. b) Calculate the other component and explain what effect it has on the sledge's contact with the ground.

A block of ice weighing $100\ \text{N}$ is on a ramp forming $30^\circ$ with the horizontal, as shown in the figure.

Calculate the component of the weight parallel to the ramp ($P \times \sen 30^\circ$) and the perpendicular component ($P \times \cos 30^\circ$), using $\sen 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.87$. Which of the two tends to make the block slide down the ramp?

A box weighing $50\ \text{N}$ is at rest on an inclined plane of $37^\circ$. Using $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$, calculate the components of the weight parallel and perpendicular to the plane.

A toy cart rests on a ramp with an adjustable angle.

1. a) Using $\sen 45^\circ \approx 0.7$ and $\cos 45^\circ \approx 0.7$, reason for which angle the component of weight parallel to the ramp is equal to the perpendicular component.
2. b) Describe what happens to the two components as the angle increases from $0^\circ$ to $90^\circ$ and explain why the cart accelerates more the steeper the ramp is.

A local municipality 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 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.

A wind force of $80\ \text{N}$ and a current force of $60\ \text{N}$ act on the sail of an Optimist dinghy, perpendicular to each other.

1. a) Calculate the magnitude of the resultant force on the boat.
2. b) Calculate the angle formed by the resultant with the direction of the wind using $\arcsen$.

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).

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 $\sen 30^\circ = 0.5$.

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.

To move a lawnmower, one can push the handle downwards and forwards, or pull it upwards and forwards. In both cases the force is $100\ \text{N}$ and forms $30^\circ$ with the horizontal ($\sen 30^\circ = 0.5$, $\cos 30^\circ \approx 0.87$).

1. a) Calculate the horizontal component in both cases and verify that it is the same.
2. b) Calculate the vertical component in each case and state its direction.
3. c) By comparing how the contact force with the ground changes (and with it, friction), explain why pulling makes it easier to move the lawnmower than pushing.

A student claims: "if two perpendicular forces of $5\ \text{N}$ each act on a body, the resultant is $10\ \text{N}$."

1. a) Explain their mistake.
2. b) Calculate the correct value of the resultant (give the answer to one decimal place).
3. c) State in which case the sum of two forces of $5\ \text{N}$ would give $10\ \text{N}$.