A Stone Is Suspended From A Point By A Piece Of String 50 Cm Long. It Swings Back And Forth. Calculate The Angle The String Makes With The Vertical When The Stone Is 35 Cm Vertically Below The Point Of Suspension.

Feb 25, 2025 by ADMIN 214 views

A Stone is Suspended from a Point by a Piece of String 50 cm Long: Calculating the Angle with the Vertical

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Introduction

In this article, we will explore the concept of simple harmonic motion and calculate the angle that a string makes with the vertical when a stone is suspended from a point by a piece of string 50 cm long and is 35 cm vertically below the point of suspension. We will use the principles of trigonometry and the properties of a right-angled triangle to solve this problem.

Understanding the Problem

A stone is suspended from a point by a piece of string 50 cm long. The stone swings back and forth, and we are interested in finding the angle that the string makes with the vertical when the stone is 35 cm vertically below the point of suspension. To solve this problem, we need to use the principles of trigonometry and the properties of a right-angled triangle.

The Right-Angled Triangle

A right-angled triangle is a triangle with one angle that is equal to 90 degrees. In this case, the string is the hypotenuse of the right-angled triangle, and the vertical distance between the stone and the point of suspension is one of the legs. The other leg is the horizontal distance between the stone and the point of suspension.

Calculating the Angle

To calculate the angle that the string makes with the vertical, we need to use the properties of a right-angled triangle. We can use the sine, cosine, or tangent function to find the angle. In this case, we will use the sine function.

The sine function is defined as:

sin(θ) = opposite side / hypotenuse

In this case, the opposite side is the vertical distance between the stone and the point of suspension, which is 35 cm. The hypotenuse is the length of the string, which is 50 cm.

Calculating the Angle Using the Sine Function

We can now calculate the angle using the sine function:

sin(θ) = 35 cm / 50 cm

sin(θ) = 0.7

To find the angle, we need to take the inverse sine of 0.7:

θ = arcsin(0.7)

θ ≈ 45.58 degrees

Calculating the Angle Using the Cosine Function

We can also calculate the angle using the cosine function:

cos(θ) = adjacent side / hypotenuse

In this case, the adjacent side is the horizontal distance between the stone and the point of suspension. We can use the Pythagorean theorem to find the horizontal distance:

a^2 + b^2 = c^2

where a is the horizontal distance, b is the vertical distance, and c is the length of the string.

a^2 + 35^2 = 50^2

a^2 + 1225 = 2500

a^2 = 1275

a ≈ 35.7 cm

Now we can calculate the angle using the cosine function:

cos(θ) = 35.7 cm / 50 cm

cos(θ) = 0.713

To find the angle, we need to take the inverse cosine of 0.713:

θ = arccos(0.713)

θ ≈ 45.58 degrees

Conclusion

In this article, we calculated the angle that a string makes with the vertical when a stone is suspended from a point by a piece of string 50 cm long and is 35 cm vertically below the point of suspension. We used the principles of trigonometry and the properties of a right-angled triangle to solve this problem. We found that the angle is approximately 45.58 degrees.

Applications

This problem has many applications in real-life situations. For example, in physics, we can use this concept to study the motion of objects in a gravitational field. In engineering, we can use this concept to design and build structures that can withstand various types of loads.

Future Work

In the future, we can explore more complex problems that involve the motion of objects in a gravitational field. We can also use this concept to study the behavior of objects in other types of fields, such as electromagnetic fields.

References

[1] "Trigonometry" by Michael Corral
[2] "Physics for Scientists and Engineers" by Paul A. Tipler
[3] "Engineering Mechanics" by Russell C. Hibbeler

Glossary

Hypotenuse: The longest side of a right-angled triangle.
Opposite side: The side of a right-angled triangle that is opposite the angle being measured.
Adjacent side: The side of a right-angled triangle that is adjacent to the angle being measured.
Sine function: A trigonometric function that is defined as the ratio of the opposite side to the hypotenuse.
Cosine function: A trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse.
Tangent function: A trigonometric function that is defined as the ratio of the opposite side to the adjacent side.

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Introduction

In our previous article, we explored the concept of simple harmonic motion and calculated the angle that a string makes with the vertical when a stone is suspended from a point by a piece of string 50 cm long and is 35 cm vertically below the point of suspension. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the relationship between the angle and the length of the string?

A: The angle is directly proportional to the length of the string. As the length of the string increases, the angle also increases.

Q: How does the angle change when the stone is moved closer to the point of suspension?

A: When the stone is moved closer to the point of suspension, the angle decreases. This is because the vertical distance between the stone and the point of suspension decreases, resulting in a smaller angle.

Q: Can we use the same method to calculate the angle when the stone is moved to a different position?

A: Yes, we can use the same method to calculate the angle when the stone is moved to a different position. We just need to update the vertical distance between the stone and the point of suspension and recalculate the angle.

Q: What is the significance of the sine function in this problem?

A: The sine function is used to calculate the angle between the string and the vertical. It is a fundamental concept in trigonometry and is used to relate the angle to the opposite side and the hypotenuse.

Q: Can we use the cosine function to calculate the angle?

A: Yes, we can use the cosine function to calculate the angle. However, we need to use the adjacent side and the hypotenuse to calculate the angle.

Q: What is the relationship between the angle and the horizontal distance?

A: The angle is directly proportional to the horizontal distance. As the horizontal distance increases, the angle also increases.

Q: Can we use the tangent function to calculate the angle?

A: Yes, we can use the tangent function to calculate the angle. However, we need to use the opposite side and the adjacent side to calculate the angle.