Question

An object of mass m has these three forces acting on it (there is no normal force, "no surface"). F1 = 5 N, F2 = 8 N, and F3 = 5 N. When answering the questions below, assume the x-direction is to the right, and the y-direction is straight upwards.
a) What is the net force in component form, in terms of F1, F2, F3, and the unit vectors i and j
b) What is the magnitude of the net force, in newtons?
c) What is the angle θ, in degrees, of the net force, measured from the +x-axis? Enter an angle between -180° and 180°.
d) What is the magnitude, |a| of the acceleration, in meters per square second, if the block has a mass of 6.9 kg?

Answer 1

The figure is missing: find it in attachment.

a) (3i - 5j) N

First of all, we have to write each force as a vector with a direction:

- Force F1 points downward, so along the negative y-direction, so we can write it as

$F_1 = -5 j$

- Force F2 points to the right, so along the positive x-direction, so we can write it as

$F_2 = +8 i$

- Force F3 points to the left, so along the negative x-direction, so we can write it as

$F_3 = -5 i$

Now we can write the net force, by adding the three vectors and separating the x-component from the y-component:

$F=F_1+F_2+F_3 = -5j+8i-5i = (8-5)i+(-5)j=(3i-5j)N$

b) 5.8 N

The magnitude of the net force can be calculated by applying Pythagorean's theorem on the components of the net force:

$F=\sqrt{F_x^2+F_y^2}$

where

$F_x = 3 N$ is the x-component

$F_y = -5 N$ is the y-component

Substituting into the equation,

$F=\sqrt{(3)^2+(-5)^2}=5.8 N$

c) $-59.0^{\circ}$

The angle of the net force, measured with respect to the positive x-axis (counterclockwise), can be calculated by using the formula

$\theta=tan^{-1} (\frac{F_y}{F_x})$

where

$F_x = 3 N$ is the x-component

$F_y = -5 N$ is the y-component

Substituting into the equation, we find

$\theta = tan^{-1} (\frac{-5}{3})=-59.0^{\circ}$

d) $0.84 m/s^2$

The acceleration can be found by using Newton's second law:

$F=ma$

where

F is the net force on an object

m is its mass

a is the acceleration

For the object in the problem, we know

F = 5.8 N

m = 6.9 kg

Solving the equation for a, we find the magnitude of the acceleration:

$a=\frac{F}{m}=\frac{5.8}{6.9}=0.84 m/s^2$