Question Video: Finding the Couple That Balances a Given Couple Acting on a Rectangle | Nagwa Question Video: Finding the Couple That Balances a Given Couple Acting on a Rectangle | Nagwa

Question Video: Finding the Couple That Balances a Given Couple Acting on a Rectangle Mathematics • Third Year of Secondary School

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𝐴𝐡𝐢𝐷 is a rectangle, where 𝐴𝐡 = 24 cm, and 𝐡𝐢 = 7 cm. Two forces, each of magnitude 43 N, are acting along 𝐡𝐴 and 𝐷𝐢, respectively. Determine the magnitude of each of the two forces acting at points 𝐡 and 𝐷 and perpendicular to the line 𝐡𝐷 that would make the whole system in equilibrium.

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Video Transcript

𝐴𝐡𝐢𝐷 is a rectangle, where the side 𝐴𝐡 equals 24 centimeters, and the side 𝐡𝐢 equals seven centimeters. Two forces, each of magnitude 43 newtons, are acting along 𝐡𝐴 and 𝐷𝐢, respectively. Determine the magnitude of each of the two forces acting at points 𝐡 and 𝐷 and perpendicular to the line 𝐡𝐷 that would make the whole system in equilibrium.

Let’s begin with a diagram of the scenario. We have the rectangle 𝐴𝐡𝐢𝐷 with side lengths 24 centimeters and seven centimeters. We have two forces of magnitude 43 newtons acting along 𝐡𝐴 and 𝐷𝐢. We have two forces, which we can call 𝐹 one and 𝐹 two, acting at the points 𝐡 and 𝐷, and perpendicular to the line 𝐡𝐷.

Without these two forces, the system is already in linear equilibrium, since the two forces of 43 newtons are antiparallel and will balance each other. Therefore, since 𝐹 one and 𝐹 two are also antiparallel, for the system to remain in linear equilibrium, 𝐹 one and 𝐹 two must balance each other as well, and so they must be the same magnitude. Let’s call this magnitude 𝐹.

This system is also in rotational equilibrium, so the couple comprising the forces 𝐹 one and 𝐹 two must be equal and opposite to the couple comprising the two forces of 43 newtons. Recall that the magnitude 𝑀 of a couple generated by two forces of magnitude 𝐹 acting perpendicularly to the ends of a line of length 𝑑 is equal to 𝐹 times 𝑑. Looking at the line 𝐡𝐢, we have two forces of 43 newtons acting from the points 𝐡 and 𝐢 and perpendicularly to the line 𝐡𝐢, which has length seven centimeters.

Therefore, the magnitude 𝑀 one of the couple comprising these two forces is 43 times seven, which is equal to 301 newton-centimeters. This couple will have equal magnitude to the couple comprising the forces 𝐹 one and 𝐹 two. This second couple, 𝑀 two, is also given by the magnitude of the unknown forces 𝐹 multiplied by the length of the line between their points of action, 𝐡𝐷.

Rearranging this equation for 𝐹 gives us 301 divided by the length of the diagonal line 𝐡𝐷, which is given via the Pythagorean theorem by the square root of the squares of the sides of the rectangle, 24 and seven. Performing this calculation gives us the magnitude of both the unknown forces, 12.04 exactly, and the unit is newtons.

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