Chapter 8: Q. 52 (page 201)

Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ωmin you must maintain if you want the ball to complete the full circle without the string going slack at the top.
a. Find an expression for ω_min
b. Evaluateω_minin rpm for a 65 g ball tied to a 1.0-m-long string.

Short Answer

Expert verified

a) The expression $ω_{m i n} = \sqrt{\frac{g}{r}}$

b) The value of ω_min = 30 rpm for the given ball.

Step by step solution

Part(a) Step 1: Given Information

mass = m
string length= L.

Part(a) Step 2 : Explanation

First draw a free body diagram as below then equate forces

At top point equate vertical force, we get

$T = (\frac{m v^{2}}{r}) - m g$

To avoid slack this force must be greater than or equal to zero

That means

$\frac{m v^{2}}{r} - m g \geq 0 \frac{v^{2}}{r} - g \geq 0 (C a n c e l i n g m) v \geq \sqrt{r g}$

Substitute v=ωr

$ω r \geq \sqrt{r g} ω \geq \sqrt{\frac{g}{r}}$

$ω_{m i n} = \sqrt{\frac{g}{r}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)$

Part(b) Step 1 : Given information

mass = 65 g ball

Length of string =1.0-m

Part(b) Step 2 : Explanation

Substitute the values in equation (1), we get

$ω_{m i n} = \sqrt{\frac{g}{r}} = \sqrt{\frac{9.8 m / s^{2}}{1 m}} = 3.132 r a d / s e c$

Convert into rpm

$ω = (3.132 r a d / s) (\frac{1}{2 π r a d}) (\frac{60 s e c}{1 m i n}) = 29.908 r p m \approx 30 r p m .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Part(a) Step 1: Given Information

Part(a) Step 2 : Explanation

Part(b) Step 1 : Given information

Part(b) Step 2 : Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Mechanics and Materials

Electric Charge Field and Potential

Classical Mechanics

Dynamics

Astrophysics

Study anywhere. Anytime. Across all devices.