Three forces act on an object, as indicated in the drawing. Force 𝐅1 has a magnitude of 21.0 new

step 1 All right, welcome back. For this problem, well, let me start off with trying my coordinate plan

Three forces act on an object, as indicated in the drawing....

Three forces act on an object, as indicated in the drawing. Force $\overrightarrow{\mathbf{F}}_{1}$ has a magnitude of 21.0 newtons $(21.0 \mathrm{N})$ and is directed $30.0^{\circ}$ to the left of the $+y$ axis. Force $\overrightarrow{\mathbf{F}}_{2}$ has a magnitude of $15.0 \mathrm{N}$ and points along the $+x$ axis. What must be the magnitude and direction (specified by the angle $\theta$ in the drawing ) of the third force $\overrightarrow{\mathbf{F}}_{3}$ such that the vector sum of the three forces is $0 \mathrm{N}$ ?

Key Concepts

Vector Addition

This concept involves combining multiple vectors to determine their net effect on an object. In force problems, vector addition is used to find the overall force acting on an object by adding the individual forces together, accounting for both their magnitudes and directions.

Resolving Vectors into Components

In many physics problems, vectors are broken down into their perpendicular (usually horizontal and vertical) components using trigonometric functions such as sine and cosine. This makes it easier to work with the vectors algebraically, especially when summing forces along each axis.

Equilibrium of Forces

This principle states that if an object is in equilibrium, the vector sum of all forces acting on it is zero. This is used to set up equations for the net forces along each axis, allowing one to solve for unknown forces that balance out the system.

Determining Magnitude and Direction

After calculating the components of a resultant force, the overall magnitude can be found using the Pythagorean theorem, and the direction is determined using inverse trigonometric functions such as the arctan function. This step is essential to fully describe the unknown force's effect.

Transcript

0:00 All right, welcome back.

00:02 For this problem, well, let me start off with trying my coordinate plane here.

00:12 Sorry, it can be a little straighter, but it'll work for our purposes, i hope.

00:17 So we've got three vectors we're doing with one, which i'll call force one.

00:23 So we're doing with forces here.

00:27 Oh, actually that's force two.

00:29 Nevermind, here's force one right here.

00:32 Force one points in this direction, forming an angle of 30 degrees with the the plus y axis and has a magnitude of 21 newtons.

00:43 So f sub 1.

00:46 All right, now we get this vector right here, f sub 2 is along the plus x axis and has a magnitude of 15 newtons.

00:57 All right, so to balance these forces out, have the net force on our object be 0.

01:03 We need to add one more vector.

01:05 You can kind of tell by looking at this, if there's this vector right here, here has a positive y component.

01:13 So our third vector needs to have a negative a negative y component.

01:21 And since both of these vectors point in opposite x directions, it will be pretty close to having no extraction, but it will be kind of, i mean, there will be a little bit probably in this direction this way, because this does appear slightly larger than the x direction.

01:41 Component of this vector.

01:43 So i will just kind of draw it out here.

01:46 This could be wrong, could be right.

01:48 We'll find out soon hopefully.

01:52 But it will look something like that i think.

01:59 All right.

02:00 So now we can do some exciting math here.

02:04 We're going to, well first of all, i don't really like this 30 degrees right here.

02:09 We're going to turn it into a 60 degree angle with the negative x -axis.

02:13 It doesn't really matter...