The Vector Nature of Forces
In a game of tug of war, two teams pull with different forces and in directions that are not always completely opposite. To predict which way the rope will move, it is not enough to add or subtract numbers: forces are vectors, and vectors are added using their own rules, which combine geometry and trigonometry.
Vector quantity: magnitude, direction and sense
If you push a piece of furniture with 100 N, it will not move in the same way if you push straight on, from the side or downwards: besides the strength, what other information do we need to know what effect it will have?
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.
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…
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.
Forces in the same direction: algebraic addition
When several forces act along the same line (even if there are more than two), the resultant is calculated by assigning a positive sign to one sense and a negative sign to the opposite one, and adding algebraically.
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.
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.
Graphical addition of forces that are not aligned: the parallelogram
If two forces do not act along the same line, their magnitudes cannot simply be added or subtracted. We use the graphical construction of the parallelogram or, equivalently, the triangle method.
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.
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,…
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.
Resolving a force into components
When you drag a suitcase by pulling on its sloping handle, part of your effort makes it move along the ground and another part lifts it upwards: how do we know what fraction of the force acts in each direction?
To add forces at any angle to each other, the most general method is the analytical one: resolve each force into its horizontal component ($F_x$) and vertical component ($F_y$).
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 $\cos 60^\circ = 0,5$).
Resultant of several forces using components
The components method is the only practical one when there are three or more forces at different angles, or when the question already gives you the components and asks you to reconstruct the magnitude.
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 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. a) Calculate the…
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.
- a) Calculate the component of the weight parallel to the ramp if it forms $15^\circ$ with the horizontal ($\sen 15^\circ \approx 0,26$).
- b) Calculate the same component for a $30^\circ$ ramp ($\sen 30^\circ = 0,5$).
- c) In light of the results, explain which design is suitable for an accessible ramp and why.
Back to the start: To find out which way the rope moves, we resolve each team's force into the horizontal and vertical axes, add their components separately ($R_x$ and $R_y$) and calculate the direction with $\tan\alpha_R = R_y / R_x$: the rope will accelerate in exactly the sense of the resultant vector.