The helicopter view in Figure P 3.31 shows two people pulling on a stubborn mule. Find (a) the s

The helicopter view in Figure P 3.31 shows two people...

The helicopter view in Figure $\mathrm{P} 3.31$ shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (symbolized N).

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to determine their overall effect. In problems involving forces, this technique is used to calculate the resultant vector by adding the corresponding components of each force vector, ensuring that both magnitude and directional contributions are accounted for.

Resultant Force

The resultant force is the single force that has the same effect as the combined individual forces acting on an object. It is determined by vector addition and represents both the overall magnitude and the direction in which the net effect of all the forces is applied.

Equilibrium

Equilibrium occurs when the total (net) force acting on an object is zero, meaning that all forces cancel each other out, and the object remains at rest or moves with a constant velocity. In problems involving multiple forces, finding the balancing force that brings the system into equilibrium is essential.

Vector Decomposition

Vector decomposition is the method of breaking a vector into its perpendicular components, usually along the horizontal and vertical axes. This technique simplifies the addition of vectors by allowing for the independent summation of each set of components, which can then be recombined to determine the resultant force.

Transcript

00:01 Hi, here in this given problem, suppose this is the symbolic representation of the mule.

00:10 Here this is the x axis and this is the y axis.

00:21 The force is being applied on the mule, force f1 at an angle 60 degree, force f2 at an angle of 75 degree as shown in the figure.

00:41 The values of these forces are f1 is equal to 120 newton and f2, this is 80 .0 newton.

00:57 Okay, in the first part of the problem we have to find a single force which should be applied on the mule equivalent to these two forces for which we use component method means we resolve these forces along x and y axis.

01:17 So, for f1 its component along x axis that is f1 x along y axis f1 y.

01:28 Similarly, for f2 this is f2 x it will be negative and here this is f2 y.

01:37 So, f1 x will be given by f1 cos 60 degree means this is 120 into cos 60 degree and it comes out to be equal to 60 newton and f1 y that is 120 sin 60 degree means it comes out to be equal to 103 .9 newton.

02:10 Then f2 x as it is along negative x axis.

02:15 So, this is minus 80 cos 75 degree comes out to be equal to minus 20 .7 newton and f2 y this is 87 sin 75 degree in positive direction in upward direction it comes out to be equal to 77 .3 newton.

02:44 So, finally, net force along x axis fx that is f1 x plus f2 x means this is 60 minus 20 .7 comes out to be equal to 39 .3 newton along positive x axis and fy that is f1 y plus f2 y.

03:15 So, this is 103 .9 plus 77 .3 comes out to be equal to 181 .2 newton...

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