Math Problem Statement
Solution
The problem you've uploaded involves a conical pendulum, where a bob moves in a horizontal circle at constant speed. Here's the breakdown:
Given Data:
- Mass of the bob:
- Length of the string:
- Circumference of the circular path:
You are tasked with finding:
- (a) The tension in the string.
- (b) The period of the motion.
Step 1: Calculate the Radius of the Circular Path
The radius of the circular path can be calculated from the given circumference: Substitute :
Step 2: Relate Geometry to Tension
The angle between the string and the vertical can be found using the geometry of the conical pendulum: where is the vertical height of the string, and . This gives:
Now find :
Step 3: Tension in the String
To calculate the tension , we consider the forces acting on the bob. The tension provides both a centripetal force and supports the weight of the bob .
We can find by solving the system of equations. The vertical component balances the weight: Substitute :
Step 4: Period of the Motion
The period can be found by using the velocity and the radius. First, calculate the velocity . Using , we solve for : Substitute :
The period is: Substitute :
Final Answers:
- Tension in the string:
- Period of motion:
Would you like further clarifications or more details on any step?
Here are 5 related questions for further exploration:
- How would the tension change if the mass of the bob were doubled?
- How does the period of motion depend on the radius of the circular path?
- What would happen to the period if the length of the string increased?
- How would the tension change if the bob moved faster in the circular path?
- What effect does gravitational acceleration have on the period?
Tip: Always break problems involving forces into their horizontal and vertical components for clarity in solving.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Circular motion
Trigonometry
Centripetal force
Newton's Second Law
Formulas
T = mg / cos(θ)
v = sqrt(T sin(θ) * r / m)
T_period = 2πr / v
r = C / (2π)
Theorems
Newton's Second Law of Motion
Centripetal force equation
Suitable Grade Level
Grades 11-12
Related Recommendation
Conical Pendulum: Tension and Period Calculation for a 0.0160 kg Bob
Conical Pendulum Problem: Calculating Speed from Tension and Angle
Find the Acceleration of a Conical Pendulum (0.1 kg Ball, 1.7 m String, 37° Angle)
Calculate the Swing Period of a Light Fixture as a Simple Pendulum
Vertical Circular Motion of 0.5 kg Object with String Tension Analysis