Vector Composition of Forces and Equilibrium of a Particle

A problem states: "the wind pushes the sail of a windsurf board with a force of $120\ \text{N}$ towards the north-east."

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 resultant force on the rope.
2. b) State which team the rope will start moving towards 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 Pythagoras' 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.

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.

The components of the force exerted by the wind on a weather vane are $F_x = -8\ \text{N}$ and $F_y = 6\ \text{N}$.

1. a) Calculate the magnitude of the resultant force.
2. b) Calculate the angle formed by the resultant with the horizontal using $\arcsen$ (note that $\arcsen 0.6 \approx 37^\circ$).

The propulsion system of a delivery drone exerts a force with components $(9,\ 12)\ \text{N}$ on it.

1. a) Calculate the magnitude of the total thrust.
2. b) Calculate the angle formed by the thrust with the horizontal using $\arcsen$ (note that $\arcsen 0.8 \approx 53^\circ$).
3. 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.

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, $\sum \vec{F} = 0$.

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 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 towards the hand holding it (that is, downwards and forwards), 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.

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 shop 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.

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.

Two elastic cords pull on the eyelet of a truck tarpaulin: one with $30\ \text{N}$ towards the east and another with $40\ \text{N}$ towards the north.

1. a) Calculate the magnitude of the resultant of these two forces.
2. b) Deduce the magnitude and direction of the force that a third cord must exert so that the eyelet is in equilibrium.
3. c) Calculate the angle formed by this third force with the westward direction (use $\arcsen 0.8 \approx 53^\circ$).

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 laboratory.

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.

On a laboratory force table, a ring is held by two cords passing over pulleys: a mass of $300\ \text{g}$ hangs from one cord in the eastward direction and a mass of $400\ \text{g}$ from the other in the northward direction. The setup is shown in the figure (top view).

Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the tension in each cord (remember that the tension equals the weight of the hanging mass).
2. b) Calculate the magnitude of the resultant of the two tensions on the ring.
3. c) Deduce what mass must be hung from a third pulley and in which direction it must be placed so that the ring remains in equilibrium at the center of the table (use $\arcsen 0.8 \approx 53^\circ$).

You pull a sledge with a rope making an angle of $37^\circ$ with the horizontal, exerting a force of $250\ \text{N}$, and the sledge 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 sledge, applying the equilibrium condition in the horizontal direction.
3. c) The sledge 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 clean-up 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 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.

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.