Q3. Three ants 1, 2 and 3 are pulling a grain with forces of magnitude 10 N, 4 N and FN as shown in the figure. Find the force F if the grain remains in equilibrium under the action of the above forces.

F = 7.21 N

Explanation:

There are two methods to find the magnitude of F.

The first method is the analytical one and the second one is the geometrical one.

The easiest is the geometrical one.

We will start with it

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● Method 1:

The geometrical method

The grain remains in equilibrum under the action of 3 forces.

The 3 forces must give if joined together a close path.

There are 3 forces so the resulting shape is a triangle. (Picture 1)

● Al Kashi theorem:

We will use Al Kashi theorem (law of cosine) to find the magnitude of F.

● F^2 = b^2 + c^2 + 2bc×cosA

● F^2 = 10^2 + 4^2 + 2×10×4 × cos(37°)

● F^2 = 52.1

● F = 7.21 N

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● method 2 (analytical one)

This method is based on projecting each vector on the x and y axis.

Since the grain remains in equilibrum the relation between the vectors is:

● Vector F + Vector 1 + Vector 2 = Null vector

Vector one is the one with 4 N magnitude and 2 is the 10 one.

Project each vector following the x-axis and the y-axis and add them together.

(Picture 2)

● x-axis

F1 is the 4 N force and F2 is the 10 N one

● Fx + F1x + F2x = 0

● Fx -4 + 10 × cos(37°) = 0

F1x is negative since it goes toward the negatives numbers. It is parallel to the x-axis that's why we have put directly the magnitude

Fx2 is not parallel to the x-axis that's why we have expressed it using the cosine relation. (Picture 3)

● Fx -4 + 10 × cos(37°) = 0

● Fx = 4 -10 × cos(37°)

● Fx = -4 after rounding to the nearest unit

● y-axis

● Fy + F1y + F2y = 0

● Fy + 0 + sin(37°) × 10 = 0

F1y equals 0 since when projecting the vector's head and tail, both were in the same point. In other words the direction of the vector was perpendicular to the y-axis.

F2y is'nt parallel to the y-axis. That's why we have expressed it using the sine.

● Fy + sin(37°)×10 =0

● Fy = -10×sin(37°)

● Fy = -6 after rounding it to the nearers unit

Notice that F, Fx and Fy are creating a right triangle.(picture 4)

● The Pythagorian theorem

● F^2 = Fx^2 + Fy^2

● F^2 = (-4)^2 + (-6)^2

● F^2 = 16+36

● F^2 = 52

● F^2 = 7.21 N

That's the same result we got using the first method.

F = 7.21 N

Explanation:

horizontal sum of forces = ∑h = 10 cos(37) - 4 = 3.98 N

vertical sum of forces = ∑v = 10 cos(90-37) = 6.02 N

therefore, the resultant F² = ∑h² + ∑v²

F = sqrt (3.98² + 6.02²)

F = 7.21 N

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