Exercises: Vector addition of forces

A problem states: "the wind pushes the sail of a windsurfing board with a force of $120\ \text{N}$ toward the northeast."

1. a) State the magnitude, direction, and sense of this force.
2. b) Explain why force is a vector quantity: what information would be lost if we only gave its magnitude?

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 $350\ \text{N}$. The situation is shown in the figure.

1. a) Calculate the magnitude of the net force on the rope.
2. b) State which team the rope will start moving toward and justify your answer.

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.

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.

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.

A painting hangs stationary on a museum wall.

1. a) What is the vector sum of all the forces acting on it? Give reasons for your answer.
2. 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 $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the weight of the flowerpot.
2. b) What force does the rope exert on the flowerpot? Justify your answer by applying the equilibrium condition for a particle, $\sum \vec{F} = 0$.

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 $\cos 60^\circ = 0.5$:

1. a) Calculate the horizontal and vertical components of the force.
2. b) State which of the two components lifts the beam.

A worker leaves a pallet of $60\ \text{kg}$ at rest on a loading ramp that forms an angle of $37^\circ$ with the horizontal, as shown in the figure.

Take $g \approx 10\ \text{m/s}^2$ and use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$.

1. a) Calculate the weight of the pallet.
2. b) Calculate the component of the weight parallel to the ramp and the component perpendicular to it.
3. c) State which of the two components tends to make the pallet slide down the ramp.

Two rescuers pull an emergency stretcher using two ropes. The forces they exert, in components, are $\vec{F}_1 = (30,\ 40)\ \text{N}$ and $\vec{F}_2 = (-30,\ 40)\ \text{N}$.

1. a) Calculate the components of the resultant force on the stretcher.
2. b) Calculate the magnitude of the resultant.
3. c) Explain why the horizontal components cancel out and what advantage this symmetrical arrangement provides for transport.

A kite flies in equilibrium held by a string pulling on it with a tension of $50\ \text{N}$ at an angle of $37^\circ$ to the horizontal, in the direction toward the hand holding it (that is, downward and forward), as shown in the figure.

Use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$.

1. a) Calculate the horizontal and vertical components of the tension on the kite.
2. b) If the kite weighs $20\ \text{N}$, calculate the vertical force that the wind must exert on it to keep it in equilibrium, applying $\sum \vec{F} = 0$ in the vertical direction.

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.

1. a) Draw the three forces head-to-tail in your notebook (polygon rule) and describe what happens to the polygon of forces.
2. b) Check numerically that $\sum \vec{F} = 0$ by summing the components along each axis.
3. c) State the mechanical state of the ring.

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

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 $\sen 30^\circ = 0.5$.

1. a) Draw the forces acting on the lamp in your notebook.
2. b) Apply the equilibrium condition in the vertical direction and calculate the tension in each cable.
3. 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.

A store sign weighing $100\ \text{N}$ hangs from a knot from which two ropes extend: one horizontal, attached to the wall, and another going up to the ceiling making an angle of $30^\circ$ with the horizontal. Use $\sen 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.87$.

1. a) Draw the force diagram acting on the knot in your notebook.
2. b) Apply the equilibrium condition in the vertical direction and calculate the tension in the inclined rope.
3. c) Apply the equilibrium condition in the horizontal direction and calculate the tension in the horizontal rope.

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

Three forces with components $\vec{F}_1 = (8,\ 6)\ \text{N}$, $\vec{F}_2 = (-3,\ 2)\ \text{N}$ and $\vec{F}_3 = (-5,\ -8)\ \text{N}$ act on the central ring of a force table in the lab.

1. a) Add the components of the three forces along each axis and check that the ring is in equilibrium.
2. b) If force $\vec{F}_3$ is suddenly removed, calculate the components of the resultant force remaining on the ring.
3. c) Predict which direction the ring will start moving in that case and justify your answer.

You pull a sled with a rope making an angle of $37^\circ$ with the horizontal, exerting a force of $250\ \text{N}$, and the sled moves across the snow at constant velocity, as shown in the figure.

Use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$.

1. a) Calculate the horizontal and vertical components of your force.
2. b) Calculate the friction force acting on the sled, applying the equilibrium condition in the horizontal direction.
3. c) The sled is moving and yet you claim it is in equilibrium. Resolve this apparent contradiction using Newton's first law.

Three volunteers pull on a stranded net during a beach cleanup day, each with a force of $120\ \text{N}$, and the three ropes form angles of $120^\circ$ with each other, as shown in the figure.

1. a) Taking one of the forces in the direction of the $x$-axis, write the components of the three forces (use $\cos 120^\circ = -0.5$ and $\sen 120^\circ \approx 0.87$).
2. b) Add the components and verify that the resultant is practically zero. Does the net move?
3. c) If one of the volunteers lets go of their rope, calculate the resultant force remaining on the net and predict what will happen.

Two tugboats pull a barge along a canal using symmetrical cables that form an angle of $37^\circ$ with the canal axis, each pulling with a force of $5000\ \text{N}$, as shown in the figure.

Use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$.

1. a) Calculate the component of each force along the canal and the resultant force on the barge.
2. b) Explain why the sideways components cancel out and what advantage having symmetrical cables provides.
3. c) If one of the tugboats lets go of its cable, calculate the sideways component that appears and predict what will happen to the trajectory of the barge.

A block of $50\ \text{kg}$ is at rest on a ramp inclined at $37^\circ$, held by a rope parallel to the ramp. Take $g \approx 10\ \text{m/s}^2$ and use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$. Friction is negligible.

1. a) Calculate the weight of the block and resolve it into components parallel and perpendicular to the ramp.
2. b) Apply the equilibrium condition in the parallel direction and calculate the tension in the rope.
3. c) Calculate the normal force exerted by the ramp. If the ramp were steeper, reason how the required tension and the normal force would change.