This question relates to the parallelogram law of vector addition, a fundamental concept in physics for finding the resultant of two forces acting at a point. The law states how to geometrically represent the resultant force when two forces are represented by the adjacent sides of a parallelogram.
Correct Option: A) The diagonal of the parallelogram drawn from the same point
If two forces of magnitudes 10N and 15N are acting at an angle of 60 degrees to each other, the magnitude of their resultant force is:
To find the magnitude of the resultant force of two forces acting at an angle to each other, we use the parallelogram law of vector addition. The formula for the magnitude of the resultant force \(R\) of two forces \(F_1\) and \(F_2\) acting at an angle \(\theta\) is given by:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \]Identify the given values:
Recall the formula for the magnitude of the resultant force:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \]Substitute the given values into the formula:
\[ R = \sqrt{(10 \text{ N})^2 + (15 \text{ N})^2 + 2(10 \text{ N})(15 \text{ N})\cos(60^\circ)} \]Calculate the squares of the forces:
\[ F_1^2 = (10 \text{ N})^2 = 100 \text{ N}^2 \] \[ F_2^2 = (15 \text{ N})^2 = 225 \text{ N}^2 \]Find the value of \(\cos(60^\circ)\):
\[ \cos(60^\circ) = 0.5 \]Substitute these values back into the resultant force equation:
\[ R = \sqrt{100 + 225 + 2(10)(15)(0.5)} \] \[ R = \sqrt{100 + 225 + 150} \]Sum the terms under the square root:
\[ R = \sqrt{475} \]Calculate the square root:
\[ R \approx 21.794 \text{ N} \]Round to one decimal place:
\[ R \approx 21.8 \text{ N} \]Correct Option: B) 21.8 N
A) 5N — This would be the magnitude if the forces were acting in opposite directions (\(\theta = 180^\circ\)), and the smaller force was subtracted from the larger one, i.e., \(|15 - 10| = 5 \text{ N}\). This is not the case here.
C) 25N — This would be the magnitude if the forces were acting in the same direction (\(\theta = 0^\circ\)), in which case the forces simply add up, i.e., \(10 + 15 = 25 \text{ N}\). This is not the case here.
D) 30N — This value is incorrect and does not correspond to any standard vector addition scenario for the given forces and angle.
If two forces of 6N and 8N act at right angles to each other, the magnitude of the resultant force is
When two forces act at right angles to each other, their resultant magnitude can be found using the Pythagorean theorem, as the forces form the two perpendicular sides of a right-angled triangle, and their resultant is the hypotenuse.
Correct Option: A) 10 N
This question relates to the parallelogram law of vector addition, a fundamental concept in physics for combining two forces acting at a point. The law states how to find the resultant of two vectors (forces in this case) both in terms of their combined effect (magnitude) and their direction.
Correct Option: C) Magnitude and direction of the resultant
The question asks for the correct statement regarding the parallelogram law of forces. This law is a fundamental principle in physics used to find the resultant of two forces acting at a point.
B) The diagonal of the parallelogram represents the resultant of two forces. This statement accurately describes the parallelogram law of forces.
To find the magnitude of the resultant force when two forces act at an angle, we use the parallelogram law of vector addition. The formula for the magnitude of the resultant force is derived from this law.
A) Equal to one of the forces. As calculated, the resultant force \(R\) is equal to \(F\), which is the magnitude of one of the individual forces.