Four masses are positioned at the corners of a rectangle, as indicated in the figure below (not to scale).

(a) Find the magnitude and direction of the net force acting on the 2.0 kg mass if x = 0.40 m and y = 0.12 m.

(b) How do your answers to part (a) change (if at all) if all sides of the rectangle are doubled in length?

The magnitude of the force will be unchanged.
The magnitude of the force will be reduced by a factor of two.
The magnitude of the force will be reduced by a factor of four.
The direction will remain unchanged.
The direction will shift clockwise.
The direction will shift counterclockwise.

To determine the net force acting on the 2.0 kg mass, we can use Newton's law of universal gravitation. The formula for the gravitational force between two masses is:

F = G * (m1 * m2) / r^2

Where:
F = gravitational force
G = gravitational constant (6.67 x 10^-11 N m^2 / kg^2)
m1 and m2 = masses of the interacting objects
r^2 = square of the distance between the centers of the masses

(a) First, let's calculate the gravitational forces between the 2.0 kg mass and each of the other masses.

For the mass at the top-left corner:
F1 = G * (2.0 kg * 5.0 kg) / (0.40 m)^2

For the mass at the top-right corner:
F2 = G * (2.0 kg * 3.0 kg) / (0.12 m)^2

For the mass at the bottom-left corner:
F3 = G * (2.0 kg * 4.0 kg) / (0.12 m)^2

For the mass at the bottom-right corner:
F4 = G * (2.0 kg * 4.0 kg) / (0.40 m)^2

Now, we can find the x-components and y-components of the net force by taking the vector sum of all the individual forces in each direction.

To find the x-components:
Net force in the x-direction = F2 * cos(θ2) - F4 * cos(θ4)
Where θ2 and θ4 are the angles between the x-axis and the lines connecting the masses.

To find the y-components:
Net force in the y-direction = F3 * cos(θ3) - F1 * cos(θ1)
Where θ1 and θ3 are the angles between the y-axis and the lines connecting the masses.

To calculate the net force magnitude and direction, we can use the Pythagorean theorem and inverse tangent function.

The magnitude of the net force is given by:
Net force magnitude = sqrt((Net force in the x-direction)^2 + (Net force in the y-direction)^2)

The direction of the net force is given by:
Net force direction = atan(Net force in the y-direction / Net force in the x-direction)

(b) If all sides of the rectangle are doubled in length, the distances between the masses will also double. This means that the gravitational forces between the masses will decrease by a factor of 4 (since the distance is squared in the formula).

However, since the distances between the masses on each side of the rectangle will all double by the same factor, the net force's x and y components will also double when calculating the net force magnitude and direction. Therefore, the magnitude of the force will still be unchanged, and the direction will remain the same.

To find the net force acting on the 2.0 kg mass, we need to calculate the individual forces exerted on it by each of the three other masses and then add them up vectorially.

(a) To find the magnitude and direction of the net force, we can use the equation for gravitational force:

F = G * (m1 * m2) / r^2

Where F is the force, G is the gravitational constant (6.67430 × 10^-11 N m^2 / kg^2), m1 and m2 are the masses, and r is the distance between them.

Here's how we can find the magnitude and direction of the net force on the 2.0 kg mass:

1. Calculate the force exerted on the 2.0 kg mass by the 3.0 kg mass:
F1 = G * (2.0 kg * 3.0 kg) / (0.4 m)^2

2. Calculate the force exerted on the 2.0 kg mass by the 4.0 kg mass:
F2 = G * (2.0 kg * 4.0 kg) / (0.12 m)^2

3. Calculate the force exerted on the 2.0 kg mass by the 5.0 kg mass:
F3 = G * (2.0 kg * 5.0 kg) / ((0.4 m)^2 + (0.12 m)^2)

4. Add the individual forces vectorially:
Net Force = F1 + F2 + F3

The magnitude of the net force can be found using the Pythagorean theorem, considering that the forces are acting at right angles to each other:

Magnitude of the net force = sqrt(F1^2 + F2^2 + F3^2)

To find the direction of the net force, you can use trigonometry as follows:

Direction of the net force = atan(F2 / F1) + atan(F3 / F1)

(b) If all sides of the rectangle are doubled in length, the distances between masses will also double. Given that the force is inversely proportional to the square of the distance, the magnitudes of the forces will be reduced by a factor of four. However, since the forces act in opposite directions, the direction of the net force will remain unchanged. Hence, the correct answer is:

The magnitude of the force will be reduced by a factor of four.
The direction will remain unchanged.

the equation of motion for an object thrown from (0,0) at an angle θ with velocity v is

y(x) = -g/(2v2 cos2θ) x2 + xtanθ

the range (where y=0 again) is

r = v2 sin2θ/g

the maximum height reached is

h = v2 sin2θ/2g

So, we know that
h = 10
θ = 60°

10 = v2 (3/4)/(2*9.8)
10 = .038 v2
v2 = 263.16
v = 16.22

The range is twice the distance to the balcony, so the balcony is at half the range:

r = 16.22 sin(120)/9.8
= 29.21 * √3/2 / 9.8
= 2.58

so, he stood 1.29m from the house