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

Expression for km

The period of oscillation of the mass is,

$T = 2 π \sqrt{\frac{m}{k}}$

Rearrange it,

$T^{2} = 4 π^{2} \frac{m}{k}$

$\frac{k}{m} = \frac{4 π^{2}}{T^{2}}$

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,

$\sum F = kΔr = {mω}^{2} r$

$ω = \sqrt{\frac{k}{m} \frac{Δr}{r}}$

Substitute all values,

$ω = \sqrt{\frac{4 π^{2}}{T^{2}} \frac{Δr}{r}}$

Calculation for rotational frequency

Frequency is,

$ω = \sqrt{\frac{4 π^{2}}{T^{2}} \frac{Δr}{r}}$

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

$ω = \sqrt{\frac{4 π^{2}}{(0.35 s)^{2}} (0.15)}$

$= 6.95 rad / s$

converted to rpm,

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

$= 66 rpm$

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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

Step by step solution

Expression for km

Expression for rotation frequency

Calculation for rotational frequency

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A mass hanging from a spring oscillates with a period of0.35s. 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

Step by step solution

Expression for km

Expression for rotation frequency

Calculation for rotational frequency

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Measurements

Energy Physics

Electricity and Magnetism

Translational Dynamics

Radiation

Study anywhere. Anytime. Across all devices.

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 $15 %$ ?