Applied Mathematics — First: Statics · Mathematics. These are 7 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 37 questions on this lesson, each with an explanation.
Two parallel forces of magnitudes 45 N and 18 N act in opposite directions. Find the magnitude of their resultant.
Correct answer: (D) 27 N
Explanation: Step 1: Opposite directions, so R = 45 − 18 = 27 N. ∴ The resultant = 27 N (in the direction of the 45 N force).
Two like parallel forces, 10 N acting at A and 30 N acting at B, are 40 cm apart (AB = 40 cm). Find the distance of the resultant from A.
Correct answer: (B) 30 cm
Explanation: Step 1: The resultant R = 10 + 30 = 40 N. Step 2: Take moments about A; only the 30 N force has a moment: 30 × 40 = 1200 N·cm. Step 3: R × x = 1200, so x = 1200 ÷ 40 = 30 cm. ∴ The resultant acts 30 cm from A.
Two unlike parallel forces act at the points A and B: the force at A is 18 N. Their resultant is 24 N and acts at the point C, where AC = 56 cm. Find the force at B and the length of AB.
Correct answer: (D) 42 N, AB = 32 cm
Explanation: Step 1: Unlike forces: R = the difference of the two forces. Since R = 24 N is greater than 18 N, the force at B is the greater one: F2 = 18 + 24 = 42 N. Step 2: The resultant acts outside AB on the side of the greater force (beyond B), in the direction of F2. Step 3: 18 × AC = 42 × BC, so BC = (18 × 56) ÷ 42 = 24 cm. Step 4: AB = AC − BC = 56 − 24 = 32 cm. ∴ F2 = 42 N and AB = 32 cm.
Three like parallel forces of 5 N, 10 N and 25 N act in the same direction at points 0, 2 m and 4 m along a line. Find the magnitude of the resultant and its distance from the 5 N force.
Correct answer: (B) 40 N, 3 m from the 5 N force
Explanation: Step 1: Same direction: R = 5 + 10 + 25 = 40 N. Step 2: Take moments about the 5 N force: 10 × 2 + 25 × 4 = 20 + 100 = 120. Step 3: x = 120 ÷ 40 = 3 m from the 5 N force. ∴ R = 40 N, 3 m from the 5 N force.
The two parallel forces F1 = 2i + j and F2 = −6i − 3j act at the points A(−2, 0) and B(2, 1) respectively. Find their resultant and the point where its line of action cuts the x-axis.
Correct answer: (A) R = −4i − 2j, at (1, 0)
Explanation: Step 1: F2 = −3F1, so the forces are parallel (unlike). R = F1 + F2 = −4i − 2j. Step 2: Moments about O: M1 = (−2)(1) − (0)(2) = −2; M2 = (2)(−3) − (1)(−6) = −6 + 6 = 0. Sum = −2k. Step 3: If R acts at (x, 0): its moment about O = (x × (−2) − 0 × (−4)) k = −2x k. Step 4: −2x = −2, so x = 1 (the line of action is x − 2y = 1). ∴ R = −4i − 2j and its line of action cuts the x-axis at (1, 0).
Two like parallel forces at the ends of a rod 40 cm long have a resultant of 60 N acting 15 cm from one end. Find the two forces (near end, then far end).
Correct answer: (C) 37.5 N and 22.5 N
Explanation: Step 1: R = P + Q = 60 N. Step 2: The resultant is 15 cm from the near end; take moments about that end: Q × 40 = 60 × 15 = 900. Step 3: Q = 900 ÷ 40 = 22.5 N, so P = 60 − 22.5 = 37.5 N. ∴ Near force = 37.5 N, far force = 22.5 N.
A, B, C, D and E lie on a straight line in this order, with AB = 5 cm, BC = 7 cm, CD = 8 cm and DE = 10 cm. Forces of 10, 40 and 50 N act at A, C and D perpendicular to AE in one direction, and forces of 20 and 60 N act at B and E in the opposite direction. Find the resultant.
Correct answer: (B) 20 N, 21 cm from A beyond A
Explanation: Step 1: Take the direction of the 10 N force as positive: R = 10 + 40 + 50 − 20 − 60 = 20 N (in the direction of the 10 N force). Step 2: Distances from A: B = 5, C = 12, D = 20, E = 30 cm. Step 3: Moments about A: 40 × 12 + 50 × 20 − 20 × 5 − 60 × 30 = 480 + 1000 − 100 − 1800 = −420. Step 4: R × x = −420, so x = −21: the negative sign puts the resultant on the other side of A from E. ∴ R = 20 N, acting 21 cm from A on EA produced beyond A.
The first 10 questions in every subject are free, each with an explanation.