Problem 108
Question
The resultant \(R\) of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is: \([2005]\) (a) \(2: 1\) (b) \(3: \sqrt{2}\) (c) \(3: 2\) (d) \(3: 2 \sqrt{2}\)
Step-by-Step Solution
Verified Answer
The ratio of larger force to smaller one is \( 3:2\sqrt{2} \) (option d).
1Step 1: Understand the Force Diagram
Let's assume we have two forces, \( F_1 \) and \( F_2 \), acting on a particle. The resultant \( R \) is at right angles to one of the forces, so it's perpendicular to \( F_1 \). We'll assume \( F_1 \) is the force to which \( R \) is perpendicular. Therefore, the relationship can be expressed as \( R = \sqrt{F_2^2 - F_1^2} \), because \( R^2 + F_1^2 = F_2^2 \) by the Pythagorean theorem.
2Step 2: Use Magnitude Condition
It's given that the magnitude of \( R \) is one third of \( F_2 \). Thus we have the equation \( R = \frac{1}{3}F_2 \). Substituting this into the Pythagorean identity, we get \( \left(\frac{F_2}{3}\right)^2 + F_1^2 = F_2^2 \).
3Step 3: Simplify the Equation
Rearrange the equation: \( \frac{F_2^2}{9} + F_1^2 = F_2^2 \). Subtract \( \frac{F_2^2}{9} \) from both sides to get \( F_1^2 = F_2^2 - \frac{F_2^2}{9} \). This simplifies to \( F_1^2 = \frac{8F_2^2}{9} \).
4Step 4: Solve for the Ratio
Taking the square root on both sides, we have \( F_1 = \frac{\sqrt{8}}{3}F_2 \). Now, \( \sqrt{8} = 2\sqrt{2} \), so \( F_1 = \frac{2\sqrt{2}}{3}F_2 \). The ratio of the larger force \( F_2 \) to the smaller force \( F_1 \) is therefore \( \frac{F_2}{F_1} = \frac{3}{2\sqrt{2}} \). Multiply numerator and denominator by \( \sqrt{2} \) to rationalize and get \( \frac{3\sqrt{2}}{4} \). However, it seems that in simplification we've missed the expressions to match options, so let's analyze this again. By rationalization, it seems \( \frac{3:2\sqrt{2}} \) matches the typical representation.
Key Concepts
Resultant ForcePythagorean TheoremMagnitude Ratio
Resultant Force
When two forces act on a particle, they can be combined into one single force called the "resultant force". Think of it as summarizing the effect of both forces into one. This resultant force often helps simplify complex problems by focusing on a single action.
In our problem, this resultant force, denoted as \( R \), is perpendicular to one of the existing forces, creating a right-angle scenario. This is important because it relates directly to the Pythagorean theorem. Visualizing the forces as vectors, these can be drawn as arrows pointing in specific directions, where the length of each arrow represents the magnitude (or strength) of the force.
When \( R \) is perpendicular, imagine the situation as a right triangle where the forces form the sides and the resultant vector forms the hypotenuse at a 90-degree angle. Thus, resultant force calculations only need basic geometry knowledge and some handy trigonometric identities to solve.
In our problem, this resultant force, denoted as \( R \), is perpendicular to one of the existing forces, creating a right-angle scenario. This is important because it relates directly to the Pythagorean theorem. Visualizing the forces as vectors, these can be drawn as arrows pointing in specific directions, where the length of each arrow represents the magnitude (or strength) of the force.
When \( R \) is perpendicular, imagine the situation as a right triangle where the forces form the sides and the resultant vector forms the hypotenuse at a 90-degree angle. Thus, resultant force calculations only need basic geometry knowledge and some handy trigonometric identities to solve.
Pythagorean Theorem
The Pythagorean theorem is central when dealing with forces that are at right angles to each other, like in this exercise. It's named after the Greek mathematician Pythagoras, and it helps us calculate the unknown side of a right triangle when we know the other two sides.
Here's how it works: in a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the theorem states:
In our scenario, because \( R \) is perpendicular, this forms a right triangle where the Pythagorean theorem could be used to relate the forces. Here, the equation becomes \( R^2 + F_1^2 = F_2^2 \), which is very useful for establishing the relation between the different forces.
Here's how it works: in a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the theorem states:
- \[ c^2 = a^2 + b^2 \]
In our scenario, because \( R \) is perpendicular, this forms a right triangle where the Pythagorean theorem could be used to relate the forces. Here, the equation becomes \( R^2 + F_1^2 = F_2^2 \), which is very useful for establishing the relation between the different forces.
Magnitude Ratio
The magnitude ratio involves understanding how the sizes of different forces compare. It's like comparing the lengths of two sides. When dealing with force vectors, you often need to find which force is stronger or larger and by what factor. This is called the magnitude ratio.
In our exercise, we want to determine the ratio of the larger force to the smaller one. Given that the magnitude of \( R \) is a fraction of one force, it hints at this ratio. Using the conditions provided in the problem, we found \( R = \frac{1}{3}F_2 \), which allows us to build equations and simplify to find out \( F_1 \).
Simplifying the ratios by rationalizing the terms often makes the ratio easier to interpret. This is why rationalization was done to convert \( \frac{3}{2\sqrt{2}} \) into a form that matches standard options, finally presenting it as \( 3:2\sqrt{2} \). This is a clean and interpretable way to understand the relationship between the magnitudes of these forces.
In our exercise, we want to determine the ratio of the larger force to the smaller one. Given that the magnitude of \( R \) is a fraction of one force, it hints at this ratio. Using the conditions provided in the problem, we found \( R = \frac{1}{3}F_2 \), which allows us to build equations and simplify to find out \( F_1 \).
Simplifying the ratios by rationalizing the terms often makes the ratio easier to interpret. This is why rationalization was done to convert \( \frac{3}{2\sqrt{2}} \) into a form that matches standard options, finally presenting it as \( 3:2\sqrt{2} \). This is a clean and interpretable way to understand the relationship between the magnitudes of these forces.
Other exercises in this chapter
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