The vector nature of forces
Two people push the same wardrobe along the floor: one towards the east, the other towards the north. The wardrobe does not move purely east or purely north: it slides diagonally. To know exactly which way and with how much force, adding numbers is not enough: we have to add vectors.
A force is a vector
You push a table forwards with 20 N and it slides across the room, but if you pull upwards with the same 20 N it barely moves: why does the same numerical value produce such different effects?
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…
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.
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…
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.
Adding forces with the same direction
When two forces act along the same line, adding them is straightforward: if they have the same sense their magnitudes are added, and if they have opposite senses they are subtracted.
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…
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.
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…
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.
Graphical addition when forces do not share a direction: the parallelogram
If the two forces do not act along the same line, they cannot simply be added or subtracted directly. We draw a parallelogram with the two forces as sides: the diagonal starting at the common origin is the resultant.
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…
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…
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 $\cos 60^\circ = 0,5$).
Numerical addition by components
A drawing helps us understand, but to calculate accurately we resolve each force into its horizontal component ($F_x$) and vertical component ($F_y$), add the components separately, and then recombine them to find the resultant.
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. a) Calculate the magnitude of…
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.
- a) Calculate the magnitude of the resultant force.
- b) Calculate the angle it forms with the horizontal 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. a)…
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.
- a) First, calculate the resultant of the two horizontal forces.
- b) Combine it with the vertical force and find the magnitude of the overall resultant.
- 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…
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 $\cos 60^\circ = 0,5$:
- a) Resolve the wind force into east and north directions.
- b) Add the components of both forces along each axis.
- c) Calculate the magnitude of the resultant force on the boat (give the answer to one decimal place).
Back to the start: If you push with $40\ \text{N}$ towards the east and $30\ \text{N}$ towards the north, the resultant force is $R = \sqrt{40^2 + 30^2} = 50\ \text{N}$ and the wardrobe will slide diagonally along the floor, making an angle of about $37^\circ$ with the east direction.