Three-way tug-of-war Three people located at A, B, and C...

Three-way tug-of-war Three people located at $A, B,$ and $C$ pull on ropes tied to a ring. Find the magnitude and direction of the force with which $C$ must pull so that no one moves (the system is in equilibrium). (FIGURE CAN'T COPY)

Key Concepts

Static Equilibrium

Static equilibrium refers to the state where the sum of all forces acting on an object is zero, meaning the object remains at rest without linear or rotational acceleration. This principle is fundamental in ensuring that, when multiple forces are applied, they balance out perfectly so the overall system does not move.

Vector Addition

Vector addition is the process of combining forces, taking into account both their magnitudes and directions. This concept allows one to calculate a resultant force by summing all individual force vectors, and in an equilibrium scenario, this resultant must be zero.

Force Resolution into Components

Force resolution involves breaking down a force into its orthogonal components, typically along the horizontal and vertical axes, using trigonometric functions. This method is essential when analyzing systems where forces are applied at angles, as it simplifies the problem into manageable parts for verifying equilibrium.

Transcript

00:01 To do this problem, we're going to need to find the horizontal and vertical components of each vector.

00:07 So first, the vertical or the horizontal components, so of a, it's going to be 100 cosine of 45 degrees, and it's going to be in the negative direction.

00:22 And then plus, now the horizontal component of this is going to be 60 times cosine of 45 degrees.

00:30 And then we're going to say that's going to be equal to the horizontal component of c.

00:37 So we'll say c and then times and then we'll have cosine of some angle theta.

00:46 Okay.

00:47 And then next we're going to find the vertical components.

00:54 So here we'll have 100 cosine of, i'm sorry, sign of 45 degrees.

01:05 And then plus 60 sign of 30 degrees, this is going to be equal to, and i'll call this c times sine of theta.

01:16 So let's calculate this here.

01:19 So cosine of 45 degrees is squared of 2 over 2.

01:24 Oh, this should be of 30 degrees here.

01:28 Okay.

01:29 Okay, so this squared of 2 over 2.

01:32 So we're going to have negative 50 square root of 2, and then cosine of 30 is going to be one half.

01:41 So we have plus 30 here.

01:43 This is going to be equal to c cosine of theta.

01:47 And then here, that sign of 45 is also going to be square root of 2 over 2.

01:51 So we have 50 square root of 2.

01:55 And then here, sine of 30 is square root of 3 over 2.

02:00 So we're going to have plus 30 square root of 3.

02:04 And this is going to be equal to c, sine.

02:07 Of theta.

02:10 So first, if we want to find the angle, well, we can divide this by this...

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