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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 蝇min 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 蝇min

b. Evaluate蝇minin rpm for a 65 g ball tied to a 1.0-m-long string.

Short Answer

Expert verified

a) The expression 蝇min=gr

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

Step by step solution

01

Part(a) Step 1: Given Information

mass = m
string length= L.

02

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=(mv2r)-mg

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

That means

mv2r-mg鈮�0v2r-g鈮�0(Cancelingm)v鈮�rg

Substitute v=蝇r

蝇r鈮�rg蝇鈮�gr

So

蝇min=gr..............................(1)

03

Part(b) Step 1  : Given information

mass = 65 g ball

Length of string =1.0-m

04

Part(b) Step 2 : Explanation

Substitute the values in equation (1), we get

蝇min=gr=9.8m/s21m=3.132rad/sec

Convert into rpm

蝇=(3.132rad/s)(12蟺rad)(60sec1min)=29.908rpm鈮�30rpm.

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