Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 69

Chapter 15, Problem 69

A 15-cm-long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Verified step by step guidance

Step 1: Identify the system as a physical pendulum, where the rod and clay together swing about a pivot point. The period of a physical pendulum is determined using the formula:

T = 2 π \sqrt \frac{I}{m g d}

, where

I

is the moment of inertia,

m

is the total mass,

g

is the acceleration due to gravity, and

d

is the distance from the pivot to the center of mass.

Step 2: Calculate the moment of inertia

I

for the system. The rod's moment of inertia about the pivot is given by

I = \frac{m L^{2}}{3}

, where

m

is the rod's mass and

L

is its length. The clay's moment of inertia is calculated as

I = m r^{2}

, where

m

is the clay's mass and

r

is the distance from the pivot.

Step 3: Determine the center of mass of the system. The center of mass is calculated using the formula:

d = m

_rodd_rod+m_clayd_claym_rod+m_clay, where

d

is the distance of each mass from the pivot.

Step 4: Combine the moments of inertia of the rod and clay to find the total moment of inertia

I

. Add the individual contributions:

I

_total=I_rod+I_clay.

Step 5: Substitute the values for

I

m

g

, and

d

into the formula for the period

T

to calculate the final result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a rod pivoted at one end, the moment of inertia can be calculated using the formula I = (1/3)ml², where m is the mass and l is the length of the rod. The clay ball's contribution to the moment of inertia must also be considered, as it is located at the end of the rod.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In the context of a pendulum, the period of oscillation is influenced by the length of the pendulum and the acceleration due to gravity. The formula for the period T of a simple pendulum is T = 2π√(I/mgh), where I is the moment of inertia, m is the total mass, g is the acceleration due to gravity, and h is the distance from the pivot to the center of mass.

Center of Mass

The center of mass is the point at which the mass of a system is concentrated and around which the mass is evenly distributed. For composite objects like the rod and clay ball, the center of mass can be found by considering the individual masses and their distances from a reference point. The position of the center of mass affects the dynamics of the pendulum, including the calculation of the period and the moment of inertia.

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Related Practice

Textbook Question

A uniform rod of mass M and length L swings as a pendulum on a pivot at distance L/4 from one end of the rod. Find an expression for the frequency f of small-angle oscillations.

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Textbook Question

A 500 g air-track glider attached to a spring with spring constant 10 N/m is sitting at rest on a frictionless air track. A 250 g glider is pushed toward it from the far end of the track at a speed of 120 cm/s. It collides with and sticks to the 500 g glider. What are the amplitude and period of the subsequent oscillations?

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Textbook Question

It has recently become possible to 'weigh' DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m³) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight—but measurable—decrease in the oscillation frequency. A vibrating cantilever of mass M can be modeled as a block of mass ⅓M attached to a spring. (The factor of ⅓ arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately—perhaps to only two significant figures—but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?

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Textbook Question

The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz. The amplitude of the head's oscillations decreases to 0.5 cm in 4.0 s. What is the head's damping constant?

Textbook Question

A 200 g oscillator in a vacuum chamber has a frequency of 2.0 Hz. 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?

Textbook Question

A block on a frictionless table is connected as shown in FIGURE P15.75 to two springs having spring constants k₁ and k₂. Find an expression for the block’s oscillation frequency f in terms of the frequencies f₁ and f₂ at which it would oscillate if attached to spring 1 or spring 2 alone.