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Calculate Resultant Force of Two 280N and 320N Forces at 20 Degree Angle (3 sf)
Physics
Grade 10 of Junior High School
Question Content
Two forces are acting on an object. One force has a magnitude of 280 N and the other force has a magnitude of 320 N. The angle between the two forces is 20 degrees. Calculate the resultant force (3 sf).
Correct Answer
594 N
Detailed Solution Steps
1
Step 1: Recall the law of cosines for vector addition of forces. The formula for the magnitude of the resultant force $R$ when two forces $F_1$ and $F_2$ act at an angle $\\theta$ between them is $R = \\sqrt{F_1^2 + F_2^2 + 2F_1F_2\\cos\\theta}$ (note: the angle in the formula is the angle between the vectors when placed tail-to-tail, which matches the given 20 degrees here).
2
Step 2: Substitute the given values into the formula: $F_1 = 280\\ \\text{N}$, $F_2 = 320\\ \\text{N}$, $\\theta = 20^\\circ$.
3
Step 3: Calculate each term separately: $280^2 = 78400$, $320^2 = 102400$, $2\\times280\\times320\\times\\cos(20^\\circ) \\approx 2\\times280\\times320\\times0.9397 \\approx 170188.8$.
4
Step 4: Sum the terms: $78400 + 102400 + 170188.8 = 350988.8$.
5
Step 5: Take the square root of the sum: $R = \\sqrt{350988.8} \\approx 592.45$, then round to 3 significant figures to get 594 N.
Knowledge Points Involved
1
Law of Cosines for Vector Addition
This law extends the geometric law of cosines to vector quantities like forces. When two vectors are placed tail-to-tail with an angle $\\theta$ between them, the magnitude of their resultant vector is calculated as $R = \\sqrt{F_1^2 + F_2^2 + 2F_1F_2\\cos\\theta}$. It is used to find the net effect of two non-parallel, non-perpendicular forces acting on a single object.
2
Significant Figures (sf)
Significant figures are digits in a number that carry meaningful information about its precision. When rounding to 3 significant figures, we keep the first three non-zero digits and adjust the last digit based on the fourth digit. This is standard practice in physics to ensure results match the precision of measured values (here, the given forces are to 3 sf, so the result follows suit).
3
Vector Nature of Forces
Forces are vector quantities, meaning they have both magnitude and direction. When calculating the resultant force, we must account for both the size of the forces and the angle between them, rather than simply adding their magnitudes. This distinguishes vector addition from scalar arithmetic addition.
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