The helicopter view in Fig. P3.15 shows two people pulling...

The helicopter view in Fig. P3.15 shows two people pulling on a stubborn mule. The person on the right pulls with a force $\overrightarrow{\mathbf{F}}_{1}$ of magnitude $120 \mathrm{~N}$ and direction of $\theta_{1}=60.0^{\circ}$ The person on the left pulls with a force $\overrightarrow{\mathbf{F}}_{2}$ of magnitude $80.0 \mathrm{~N}$ and direction of $\theta_{2}=75.0^{\circ} .$ 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 $\mathrm{N}$ ).

Key Concepts

Equilibrium

Equilibrium refers to the state in which the net force acting on an object is zero, meaning that all the individual forces balance each other out. In such cases, any additional force applied is equal in magnitude and opposite in direction to the resultant force, maintaining the object’s state of rest or constant velocity.

Resultant Force

The resultant force is the single force that has the same effect as the combination of two or more forces acting on an object. It is calculated by summing the individual force components and then determining the final magnitude and direction, representing the overall effect of all contributing forces.

Resolving Forces into Components

Resolving forces into components involves breaking down a vector into perpendicular directions, typically horizontal and vertical components, using trigonometric functions like sine and cosine. This method simplifies the addition of vectors by allowing each component to be combined separately along the common axes.

Vector Addition

Vector addition is the process of combining two or more vectors (such as forces, velocities, or displacements) into a single resultant vector. This technique involves considering both the magnitude and the direction of each vector, and it is essential for accurately analyzing systems where forces act at various angles.

Transcript

00:01 Hi, here in this given problem this is the symbolic representation of the mule, a stubborn mule to show it we draw a cartesian plane having x axis and y axis.

00:20 First force suppose the mule is at the origin, one of the force being applied at the mule that is f 1, f 2 at an angle theta 1.

00:34 So, this f 1 is given as 120 newton and its angle with the positive x axis theta 1 that is 60 .0 degree.

00:46 Then another force this is f 2 and its magnitude is 80 .0 newton and it is making an angle theta 2 is equal to 75 .0 degree with the x axis negative x axis.

01:09 So, the angle between these two forces if we assume it to be given by theta in the first part of the problem where we have to find resultant of the two forces.

01:23 So, to find its angle between them it will be given by 180 degree total angle minus theta 1 plus theta 2 means this is 180 degree minus 60 degree plus 75 degree and it is calculated to be equal to 45 degree.

01:47 So, the resultant force of these two forces using triangle law of vector addition it will be given by f is equal to square root of f 1 square plus f 2 square plus twice of f 1 f 2 cos theta angle between them.

02:07 Plugging in the known values f will be given by square root of square of 120 plus square of 80 plus 2 times of 120 multiplied by 80 multiplied by cosine of angle between them which was 45 degree.

02:30 So, the magnitude of this resultant force that is calculated to be equal to 185 .4 newton.

02:36 One of the answer for this given problem here in this first part of the problem.

02:42 Now, if we want to find its direction also.

02:46 So, if we complete a parallelogram like this then the resultant will be given by this diagonal of the parallelogram...

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