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In this lesson, we will learn how to use the equivalence of two couples to solve different problems.

Q1:

π΄ π΅ is a horizontal light rod having a length of 60 cm, where two forces, each of magnitude 45 N, are acting vertically at π΄ and π΅ in two opposite directions. Two other forces, each of magnitude 120 N, are acting in two opposite directions at points πΆ and π· of the rod, where πΆ π· = 4 5 c m . If they form a couple equivalent to the couple formed by the first two forces, find the measure of the angle of inclination that the second two forces make with the rod.

Q2:

In a rectangle π΄ π΅ πΆ π· , π΄ π΅ = 1 8 c m and π΅ πΆ = 2 4 c m . Two forces, each of magnitude 360 N, are acting along ο« π΄ π΅ and ο« πΆ π· . Two other forces, each of magnitude πΉ , are parallel to β ο© ο© ο© ο© β π΅ π· and acting on the points π΄ and πΆ . If the two couples are equivalent, find the value of πΉ .

Q3:

π΄ π΅ πΆ π· is a parallelogram in which π β π΄ = 6 0 β , π΄ π΅ = 8 c m , and the diagonal π΅ π· is perpendicular to π΄ π΅ . If two forces, each of magnitude 7 N, are acting along ο« π΅ π΄ and ο« π· πΆ , find the magnitude of each of the two forces F and β F that act at π΄ and πΆ perpendicularly to the diagonal π΄ πΆ so that they form a couple equivalent to the couple by the two forces mentioned earlier.

Q4:

π΄ π΅ πΆ π· is a rectangle, where π΄ π΅ = 8 c m and π β π΄ π· π΅ = 1 5 β . Two forces of the same magnitude 8 newtons are acting along ο« π΄ π΅ and ο« πΆ π· , and another two forces of the same magnitude πΉ form a couple that is acting at π΅ and π· , where one of them makes an angle of 3 0 β with β ο© ο© ο© ο© β π΅ πΆ . Determine the magnitude of πΉ so that the couple formed by the last two forces is equivalent to that formed by the first two forces.

Q5:

As shown in the figure, π΄ π΅ πΆ π· is a square of side length 1 m. Two forces, each of magnitude 16 kg-wt, are acting along ο« π΄ π΅ and ο« πΆ π· , and another two forces, each of magnitude πΉ , are acting at π΅ and π· , where one of them makes an angle of 1 5 β with οͺ π΅ πΆ , and the other makes an angle of 1 5 β with ο« π· π΄ . If the couple formed by the first two forces is equivalent to that formed by the other two, determine the magnitude of πΉ and round to two decimal places.

Q6:

π΄ π΅ is a rod having a length of 6 m and negligible weight. Two equal forces, perpendicular to the rod and each with a magnitude of 70 N , are acting at its points of trisection in opposite directions. Given that these two forces are replaced by two other forces, each with a magnitude of 170 N , that are acting at the ends of the rod, such that they form a couple that is equivalent to the first, determine the angle of inclination of these two forces with the rod rounding the result to the nearest minutes.

Q7:

π΄ π΅ πΆ π· is a rectangle, where π΄ π΅ = 3 2 c m , and π β π΄ π· π΅ = 3 0 β . Two forces, each with a magnitude of 16 newtons act along ο« π΄ π΅ and ο« πΆ π· , forming a couple. If, instead of these forces, two other forces, each with magnitude πΉ newtons, were to act outside the rectangle on π΅ and π· such that they made angles of 1 5 β with π΅ πΆ and π· π΄ , respectively and formed an equivalent couple to the first two forces, find the value of πΉ .

Q8:

In a rectangle π΄ π΅ πΆ π· , π΄ π΅ = 7 2 c m and π΅ πΆ = 9 6 c m . Two forces, each of magnitude 930 N, are acting along ο« π΄ π΅ and ο« πΆ π· . Two other forces, each of magnitude πΉ , are parallel to β ο© ο© ο© ο© β π΅ π· and acting on the points π΄ and πΆ . If the two couples are equivalent, find the value of πΉ .

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