Vector composition of forces

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 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.

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.

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 $\arcsen 0,6 \approx 37^\circ$).

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.

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

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 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.