Vector Addition of Forces
Two tugboats pull a barge up the Mississippi River with cables that are not parallel; a crane cable holds a beam at some angle; three people pull on a stranded net from different directions. In none of these cases is it enough to add or subtract numbers: forces are vectors, and to predict what will happen to the body we have to add them using the rules of vector addition.
Force: magnitude and direction
If you push a shopping cart forward with $50\ \text{N}$ it moves, but if you apply the same $50\ \text{N}$ downwards you just squash it against the floor: why is the numerical value not enough to describe what a force does?
A problem states: "the wind pushes the sail of a windsurfing board with a force of $120\ \text{N}$ toward the northeast." a) State the magnitude, direction, and sense of…
A problem states: "the wind pushes the sail of a windsurfing board with a force of $120\ \text{N}$ toward the northeast."
- a) State the magnitude, direction, and sense of this force.
- b) Explain why force is a vector quantity: what information would be lost if we only gave its magnitude?
Forces in the same direction and graphical addition: the parallelogram
When forces act along the same line, the net force is an algebraic sum with signs. When they are not aligned, here is a quick review of the graphical parallelogram construction; the main tool in this course is components, in the next section.
The two auxiliary thrusters of a space probe ignite simultaneously and push in the same direction and sense with forces of $250\ \text{N}$ and $400\ \text{N}$. Calculate…
The two auxiliary thrusters of a space probe ignite simultaneously and push in the same direction and sense with forces of $250\ \text{N}$ and $400\ \text{N}$. Calculate the resultant force on the probe and draw a diagram in your notebook representing the two forces and the resultant.
In a tug-of-war game at a local festival, the home team pulls to the left with a force of $380\ \text{N}$ and the visiting team pulls to the right with $350\ \text{N}$.…
In a tug-of-war game at a local festival, the home team pulls to the left with a force of $380\ \text{N}$ and the visiting team pulls to the right with $350\ \text{N}$. The situation is shown in the figure.
- a) Calculate the magnitude of the net force on the rope.
- b) State which team the rope will start moving toward and justify your answer.
Two forces of $4\ \text{N}$ and $3\ \text{N}$ act on the same point at an angle of $90^\circ$ to each other. Draw the situation in your notebook using a scale of…
Two forces of $4\ \text{N}$ and $3\ \text{N}$ act on the same point at an angle of $90^\circ$ to each other. Draw the situation in your notebook using a scale of $1\ \text{cm}$ per newton, apply the parallelogram rule and measure the diagonal to obtain the magnitude of the resultant. Then check the result with the Pythagorean theorem and compare both values.
You can do it on paper and send a photo with your answer.
Resolving a force into components
Weight comes first: the weight of an object of mass $m$ is $F_g = mg$, straight down, with $g = 9.8\ \text{m/s}^2$ (for example, a 6 kg object weighs about $59\ \text{N}$). Some problems tell you to use $g \approx 10\ \text{m/s}^2$ to keep the arithmetic simple; when a problem gives a value of $g$, use that one, and otherwise use $9.8\ \text{m/s}^2$. To add forces at any angle to each other, the general method is the analytical one: resolve each force into its horizontal component ($F_x$) and vertical component ($F_y$).
Two people pull a moving crate with ropes perpendicular to each other, one with a force of $60\ \text{N}$ and the other with $80\ \text{N}$. Draw the situation in your…
Two people pull a moving crate with ropes perpendicular to each other, one with a force of $60\ \text{N}$ and the other with $80\ \text{N}$. Draw the situation in your notebook and calculate the magnitude of the resultant force on the crate using the Pythagorean theorem.
You can do it on paper and send a photo with your answer.
The two motors of a competition robot push such that the components of the total force on the robot are $F_x = 12\ \text{N}$ and $F_y = 5\ \text{N}$. Calculate the…
The two motors of a competition robot push such that the components of the total force on the robot are $F_x = 12\ \text{N}$ and $F_y = 5\ \text{N}$. Calculate the magnitude of the resultant force.
The cable of a construction crane pulls a beam with a force of $400\ \text{N}$ making an angle of $60^\circ$ with the horizontal, as shown in the figure. Using…
The cable of a construction crane pulls a beam with a force of $400\ \text{N}$ making an angle of $60^\circ$ with the horizontal, as shown in the figure.
Using $\sen 60^\circ \approx 0.87$ and $\cos 60^\circ = 0.5$:
- a) Calculate the horizontal and vertical components of the force.
- b) State which of the two components lifts the beam.
Resultant of several forces using components
The components method is essential when there are three or more forces at different angles, or when the question already gives the components and asks you to reconstruct the magnitude and the angle.
To add more than two forces graphically, use the head-to-tail (polygon) method: draw the first force, start the second at the tip of the first, and so on. The net force is the arrow from the start of the first to the tip of the last. If the polygon closes on itself, the net force is zero. To get the angle from the components, use $\tan\theta = R_y/R_x$, or $\sin\theta = R_y/R$ with $\sin^{-1}$ (arcsin) on your calculator, and check the signs of $R_x$ and $R_y$ to place the angle in the right quadrant.
The components of the force exerted by the wind on a weather vane are $F_x = -8\ \text{N}$ and $F_y = 6\ \text{N}$. a) Calculate the magnitude of the resultant force. b)…
The components of the force exerted by the wind on a weather vane are $F_x = -8\ \text{N}$ and $F_y = 6\ \text{N}$.
- a) Calculate the magnitude of the resultant force.
- b) Calculate the angle formed by the resultant with the horizontal using $\arcsen$ (note that $\arcsen 0.6 \approx 37^\circ$).
Three forces with components $\vec{F}1 = (6,\ 0)\ \text{N}$, $\vec{F}2 = (0,\ 8)\ \text{N}$ and $\vec{F}3 = (-6,\ -8)\ \text{N}$ act on a ring. a) Draw the three forces…
Three forces with components $\vec{F}_1 = (6,\ 0)\ \text{N}$, $\vec{F}_2 = (0,\ 8)\ \text{N}$ and $\vec{F}_3 = (-6,\ -8)\ \text{N}$ act on a ring.
- a) Draw the three forces head-to-tail in your notebook (polygon rule) and describe what happens to the polygon of forces.
- b) Check numerically that $\sum \vec{F} = 0$ by summing the components along each axis.
- c) State the mechanical state of the ring.
You can do it on paper and send a photo with your answer.
The propulsion system of a delivery drone exerts a force with components $(9,\ 12)\ \text{N}$ on it. a) Calculate the magnitude of the total thrust. b) Calculate the…
The propulsion system of a delivery drone exerts a force with components $(9,\ 12)\ \text{N}$ on it.
- a) Calculate the magnitude of the total thrust.
- b) Calculate the angle formed by the thrust with the horizontal using $\arcsen$ (note that $\arcsen 0.8 \approx 53^\circ$).
- c) Draw the thrust vector and its two components in your notebook.
You can do it on paper and send a photo with your answer.
Equilibrium of a particle
In a tug-of-war the central flag stays still if both teams pull with the same force, but if several cables pull in different directions, what exact mathematical condition predicts whether or not the object will start to move?
Two facts you will use all the time. First, an object hanging at rest from a rope means the tension equals its weight: $T = F_g = mg$. Second, if some forces do not add to zero, the single extra force that would bring the particle to equilibrium (the equilibrant) has the same magnitude as the net force and points the opposite way: $F_{eq,x} = -\sum F_x$ and $F_{eq,y} = -\sum F_y$. Find its magnitude with Pythagoras and its angle with $\sin^{-1}$ or $\tan^{-1}$ of the components.
A painting hangs stationary on a museum wall. a) What is the vector sum of all the forces acting on it? Give reasons for your answer. b) What would happen to the…
A painting hangs stationary on a museum wall.
- a) What is the vector sum of all the forces acting on it? Give reasons for your answer.
- b) What would happen to the painting if this sum were no longer zero?
A flowerpot of $1.2\ \text{kg}$ hangs at rest from a rope on the porch of a house. Take $g \approx 10\ \text{m/s}^2$. a) Calculate the weight of the flowerpot. b) What…
A flowerpot of $1.2\ \text{kg}$ hangs at rest from a rope on the porch of a house. Take $g \approx 10\ \text{m/s}^2$.
- a) Calculate the weight of the flowerpot.
- b) What force does the rope exert on the flowerpot? Justify your answer by applying the equilibrium condition for a particle, $\sum \vec{F} = 0$.
A lamp of $4\ \text{kg}$ hangs in equilibrium from the ceiling of a porch, supported by two symmetrical cables that make an angle of $30^\circ$ with the ceiling, as…
A lamp of $4\ \text{kg}$ hangs in equilibrium from the ceiling of a porch, supported by two symmetrical cables that make an angle of $30^\circ$ with the ceiling, as shown in the figure.
Take $g \approx 10\ \text{m/s}^2$ and use $\sen 30^\circ = 0.5$.
- a) Draw the forces acting on the lamp in your notebook.
- b) Apply the equilibrium condition in the vertical direction and calculate the tension in each cable.
- c) Compare the tension with the weight of the lamp and explain what would happen to the tensions if the angle with the ceiling were smaller. Could the cables ever be perfectly horizontal? Justify your answer.
You can do it on paper and send a photo with your answer.
Back to the start: If the two Mississippi tugboats each pull with $3{,}000\ \text{N}$ and the angle between their cables is $60^\circ$ ($30^\circ$ to the line of travel), the useful forward force is not $6{,}000\ \text{N}$ but $2 \times 3{,}000 \times \cos 30^\circ \approx 5{,}196\ \text{N}$.