Explanation Given information: The period ( τ ) of its small oscillation is observed to be 1.03 sec. The distance ( r a ) is 6 in. The mass of the rod is m . The acceleration due to gravity is g . Calculation: Convert the unit of the distance r a from inches to feet. r a = 6 in . = 6 in . × 1 ft 12 in . = 0.5 ft Calculate the time period ( τ ) of oscillation: τ = 2 π ω n Here, ω n is the natural circular frequency. Substitute 1.03 s for τ . 1.03 s = 2 π ω n ω n = 2 π 1.03 ω n = 6.1002 rad / s At maximum displacement position: The kinetic energy corresponding to the maximum displacement position is zero. In the connecting rod, the displacement of the center of gravity when it is displaced about an angle θ m is r a ( 1 − cos θ m ) . Here, the distance at which the center of gravity is located is r a . Express the maximum displacement x m as: x m = r a ( 1 − cos θ m ) As the angular displacement of the rod ( θ m ) is very small, take the value of ( 1 − cos θ m ) as θ m 2 / 2 . Thus, the maximum displacement of the rod is 1 / 2 r a θ m 2 . Write the expression potential energy ( V 1 ) corresponding to the maximum displacement position: V 1 = m g x m Substitute 1 / 2 r a θ m 2 for x m . V 1 = m g x m = m g ( 1 / 2 r a θ m 2 ) = 1 2 m g r a θ m 2 (1) At maximum velocity position: The potential energy corresponding to the maximum velocity position is zero. Write the expression for the kinetic energy ( T 2 ) corresponding to the maximum displacement position: T 2 = 1 2 m v G 2 + 1 2 I ¯ θ ˙ m 2 (2) Here, v G is velocity of the center of gravity, m is the mass of the rod , I ¯ is the moment of inertia of the rod, and θ ˙ m the derivative of the angular displacement. Write the expression for the velocity ( v G ) of the center of gravity: v G = r a θ ˙ m Write the expression for the moment of inertia of the rod: I ¯ = m k ¯ 2 Here, k ¯ 2 is the radius of gyration of the connecting rod. Substitute r a θ ˙ m for v G and m k ¯ 2 for I ¯ in Equation (2). T 2 = 1 2 m v G 2 + 1 2 I ¯ θ ˙ m 2 = 1 2 m ( r a θ ˙ m ) 2 + 1 2 ( m k ¯ 2 ) θ ˙ m 2 = 1 2 m ( r a 2 + k ¯ 2 ) θ ˙ m 2 (3) Assume the resulting motion of the rod as simple harmonic motion

11th Edition

ISBN: 9781259633133

Author: BEER

Publisher: MCG

See similar textbooks

Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

Videos

Question

Chapter 19.3, Problem 19.74P

To determine

The centroidal radius of gyration (k¯) of the connecting rod.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.3, Problem 19.73P

Chapter 19.3, Problem 19.75P

Students have asked these similar questions

L q=8 kN/m P= 12 kN En stång belastas av en punklast P vid sin ena ände samt av en jämnt utbredd last q längs hela sin längd. Stången har en tvärsnittsarea A = 150 mm² och är tillverkad av stål med elasticitetsmodul E-210 GPa. Stångens längd, i sitt obelastade tillstånd, är Z-3 m. a) Hur stor är den största normalspänning som uppstår i stången? b) Hur stor blir förlängningen av stången, orsakad av lasterna P och q?

A turbocharged engine with a compression ratio of 8 is being designed using an air standard cycle. The ambient air is assumed to be 300K and 100 kPa. The temperature at the end of the compression in the cylinder is desired to be 1000K, assuming no combustion prior to reaching TDC. At the end of the cylinder expansion the temperature is also desired to be 1000K. If both the turbine and the compressor have mechanical efficiencies of 80%, what will be the pressure ratio of the compressor and what will be the turbine exhaust temperature?

Q6: A turbocharged engine with a compression ratio of 8 is being designed using an air standard cycle. The ambient air is assumed to be 300K and 100 kPa. The temperature at the end of the compression in the cylinder is desired to be 1000K, assuming no combustion prior to reaching TDC. At the end of the cylinder expansion the temperature is also desired to be 1000K. If both the turbine and the compressor have mechanical efficiencies of 80%, what will be the pressure ratio of the compressor and what will be the turbine exhaust temperature?

Chapter 19 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The centroidal radius of gyration ( k ¯ ) of the connecting rod.

Solution Summary: The author explains the centroidal radius of gyration of the connecting rod. The period of its small oscillation is observed to be 1.03 sec.

Concept explainers

Videos

The centroidal radius of gyration (k¯) of the connecting rod.

Want to see the full answer?

Chapter 19 Solutions