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Chapter 15: Q. 50 (page 417)

A mass hanging from a spring oscillates with a period of $0.35 s$ . Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the spring's length to stretch by $15 %$ ?

Short Answer

Expert verified

Rotational frequency in rpm is $66 rpm$ .

Step by step solution

01

Expression for km

The period of oscillation of the mass is,

$T = 2 蟺 \sqrt{\frac{m}{k}}$

Rearrange it,

$T^{2} = 4 蟺^{2} \frac{m}{k}$

$\frac{k}{m} = \frac{4 蟺^{2}}{T^{2}}$

02

Expression for rotation frequency

A radial force acting toward the path's center of curvature is required for the mass to travel in a uniform circular motion.

In this situation, the radial force is the force exerted by the spring on the mass.

The net force on the mass is,

$鈭� F = 办螖谤 = {尘蝇}^{2} r$

$蝇 = \sqrt{\frac{k}{m} \frac{螖谤}{r}}$

Substitute all values,

$蝇 = \sqrt{\frac{4 蟺^{2}}{T^{2}} \frac{螖谤}{r}}$

03

Calculation for rotational frequency

Frequency is,

$蝇 = \sqrt{\frac{4 蟺^{2}}{T^{2}} \frac{螖谤}{r}}$

Where $T = 0.35 s$ , $\frac{鈭� r}{r} = 0.15$

$蝇 = \sqrt{\frac{4 蟺^{2}}{(0.35 s)^{2}} (0.15)}$

$= 6.95 rad / s$

converted to rpm,

$= (6.95 rad / s) (\frac{1 revolution}{2 蟺谤补诲}) (\frac{60 s}{1 \min})$

$= 66 rpm$

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Most popular questions from this chapter

A block on a frictionless table is connected as shown in FIGURE P15.75 to two springs having spring constants $k 1$ and $k 2$ . Find an expression for the block鈥檚 oscillation frequency $f$ in terms of the frequencies $f 1$ and $f 2$ at which it would oscillate if attached to spring $1$ or spring $2$ alone.

A $200 g$ oscillator in a vacuum chamber has a frequency of $2.0 H z$ . When air is admitted, the oscillation decreases to $60 %$ of its initial amplitude in $50 s$ . How many oscillations will have been completed when the amplitude is $30 %$ of its initial value?

Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6Hz . What was the moment of inertia of the lower leg about the knee joint?

It is said that Galileo discovered a basic principle of the pendulum鈥�

that the period is independent of the amplitude鈥攂y using

his pulse to time the period of swinging lamps in the cathedral

as they swayed in the breeze. Suppose that one oscillation of a

swinging lamp takes $5.5 s$ .

a. How long is the lamp chain?

b. What maximum speed does the lamp have if its maximum

angle from vertical is ${3.0}^{鈭�}$ ?

A block attached to a spring with unknown spring constant oscillates with a period of $2.0 s$ .What is the period if

a. The mass is doubled?

b. The mass is halved?

c. The amplitude is doubled?

d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

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