
Concept explainers
A slender rod of length l and mass m is pivoted about a point C located at a distance b from its center G. It is released from rest in a horizontal position and swings freely. Determine (a) the angular velocity of the rod as it passes through a vertical position if b = l/2, (b) the distance b for which the angular velocity of the rod as it passes through a vertical position is maximum, (c) the corresponding values of its angular velocity and of the reaction at C using the value of b calculated.
Fig. P17.16
(a)

Find the angular velocity of the rod when
Answer to Problem 17.16P
The angular velocity of the rod when
Explanation of Solution
Show the free-body diagram of the given condition as in Figure 1.
Find the mass moment of inertia of the slender rod
Here, the mass of the slender rod is m and the length of the slender rod is l.
Position 1 (Horizontal position):
The angular velocity
The velocity
Find the total kinetic energy in the horizontal position
Substitute 0 for
Positon 2 (Vertical position):
Find the velocity of the slender rod
Find the total kinetic energy in the vertical position
Substitute
Find the work done
Here, the acceleration due to gravity is g.
Write the equation of work and energy for the system using the equation.
Substitute 0 for
Therefore, the angular velocity of the rod when
(b)

Find the distance b for which the angular velocity of rod as it passes through a vertical position is maximum.
Answer to Problem 17.16P
The distance b for which the angular velocity of the rod is maximum in vertical position is
Explanation of Solution
Position 1 (Horizontal position):
Show the free-body diagram of the horizontal position as in Figure 2.
Find the mass moment of inertia of the slender rod
The angular velocity
The velocity
Find the total kinetic energy in the horizontal position
Substitute 0 for
The elevation (h) of the pivot C is zero.
Find the total potential energy
Substitute 0 for h.
Positon 2 (Vertical position):
Show the free-body diagram of the vertical position as in Figure 3.
Find the velocity of the slender rod
Find the total kinetic energy in the vertical position
Substitute
The elevation of the pivot C is
Find the total potential energy
Substitute b for h.
Write the equation of conservation of energy using the equation.
Substitute 0 for
Integrate the angular velocity with respect to b and equate to zero.
Therefore, the distance b for which the angular velocity of the rod is maximum in vertical position is
(c)

Find the angular velocity where the vertical position is maximum and the reaction at pivot C.
Answer to Problem 17.16P
The angular velocity corresponding to the maximum vertical position is
The reaction at pivot C is
Explanation of Solution
Refer to the calculation of part (b):
Substitute
Therefore, the angular velocity corresponding to the maximum vertical position is
Show the free-body diagram of the slender rod as in Figure 4.
Find the normal acceleration
Substitute
The value of tangential acceleration is
Resolve the vertical component of forces.
Take moment about point C as follows;
Therefore,
Resolve the horizontal component of forces.
Find the resultant reaction at point C using the relation.
Substitute 0 for
Therefore, the reaction at pivot C is
Want to see more full solutions like this?
Chapter 17 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
Additional Engineering Textbook Solutions
Automotive Technology: Principles, Diagnosis, And Service (6th Edition) (halderman Automotive Series)
BASIC BIOMECHANICS
Database Concepts (8th Edition)
Concepts Of Programming Languages
Mechanics of Materials (10th Edition)
Thermodynamics: An Engineering Approach
- (I) [40 Points] Using centered finite difference approximations as done in class, solve the equation for O: d20 dx² + 0.010+ Q=0 subject to the boundary conditions shown in the stencil below. Do this for two values of Q: (a) Q = 0.3, and (b) Q= √(0.5 + 2x)e-sinx (cos(5x)+x-0.5√1.006-x| + e −43*|1+.001+x* | * sin (1.5 − x) + (cosx+0.001 + ex-1250+ sin (1-0.9x)|) * x - 4.68x4. For Case (a) (that is, Q = 0.3), use the stencil in Fig. 1. For Case (b), calculate with both the stencils in Fig. 1 and Fig 2. For all the three cases, show a table as well as a plot of O versus x. Discuss your results. Use MATLAB and hand in the MATLAB codes. 1 0=0 x=0 2 3 4 0=1 x=1 Fig 1 1 2 3 4 5 6 7 8 9 10 11 0=0 x=0 0=1 x=1 Fig 2arrow_forwardFig 2 (II) [60 Points] Using centered finite difference approximation as done in class, solve the equation: 020 020 + მx2 მy2 +0.0150+Q=0 subject to the boundary conditions shown in the stencils below. Do this for two values of Q: (a) Q = 0.3, and (b) Q = 10.5x² + 1.26 * 1.5 x 0.002 0.008. For Case (a) (that is, Q = 0.3) use Fig 3. For Case (b), use both Fig. 3 and Fig 4. For all the three cases, show a table as well as the contour plots of versus (x, y), and the (x, y) heat flux values at all the nodes on the boundaries x = 1 and y = 1. Discuss your results. Use MATLAB and hand in the MATLAB codes. (Note that the domain is (x, y)e[0,1] x [0,1].) 0=0 0=0 4 8 12 16 10 Ꮎ0 15 25 9 14 19 24 3 11 15 0=0 8-0 0=0 3 8 13 18 23 2 6 сл 5 0=0 10 14 6 12 17 22 1 6 11 16 21 13 e=0 Fig 3 Fig 4 Textbook: Numerical Methods for Engineers, Steven C. Chapra and Raymond P. Canale, McGraw-Hill, Eighth Edition (2021).arrow_forwardShip construction question. Sketch and describe the forward arrangements of a ship. Include componets of the structure and a explanation of each part/ term. Ive attached a general fore end arrangement. Simplfy construction and give a brief describion of the terms.arrow_forward
- Problem 1 Consider R has a functional relationship with variables in the form R = K xq xx using show that n ✓ - (OR 1.) = i=1 2 Их Ux2 Ихэ 2 (177)² = ² (1)² + b² (12)² + c² (1)² 2 UR R x2 x3arrow_forward4. Figure 3 shows a crank loaded by a force F = 1000 N and Mx = 40 Nm. a. Draw a free-body diagram of arm 2 showing the values of all forces, moments, and torques that act due to force F. Label the directions of the coordinate axes on this diagram. b. Draw a free-body diagram of arm 2 showing the values of all forces, moments, and torques that act due to moment Mr. Label the directions of the coordinate axes on this diagram. Draw a free body diagram of the wall plane showing all the forces, torques, and moments acting there. d. Locate a stress element on the top surface of the shaft at A and calculate all the stress components that act upon this element. e. Determine the principal stresses and maximum shear stresses at this point at A.arrow_forward3. Given a heat treated 6061 aluminum, solid, elliptical column with 200 mm length, 200 N concentric load, and a safety factor of 1.2, design a suitable column if its boundary conditions are fixed-free and the ratio of major to minor axis is 2.5:1. (Use AISC recommended values and round the ellipse dimensions so that both axes are whole millimeters in the correct 2.5:1 ratio.)arrow_forward
- 1. A simply supported shaft is shown in Figure 1 with w₁ = 25 N/cm and M = 20 N cm. Use singularity functions to determine the reactions at the supports. Assume El = 1000 kN cm². Wo M 0 10 20 30 40 50 60 70 80 90 100 110 cm Figure 1 - Problem 1arrow_forwardPlease AnswerSteam enters a nozzle at 400°C and 800 kPa with a velocity of 10 m/s and leaves at 375°C and 400 kPa while losing heat at a rate of 26.5 kW. For an inlet area of 800 cm2, determine the velocity and the volume flow rate of the steam at the nozzle exit. Use steam tables. The velocity of the steam at the nozzle exit is m/s. The volume flow rate of the steam at the nozzle exit is m3/s.arrow_forward2. A support hook was formed from a rectangular bar. Find the stresses at the inner and outer surfaces at sections just above and just below O-B. -210 mm 120 mm 160 mm 400 N B thickness 8 mm = Figure 2 - Problem 2arrow_forward
- Steam flows steadily through a turbine at a rate of 45,000 lbm/h, entering at 1000 psia and 900°F and leaving at 5 psia as saturated vapor. If the power generated by the turbine is 4.1 MW, determine the rate of heat loss from the steam. The enthalpies are h1 = 1448.6 Btu/lbm and h2 = 1130.7 Btu/lbm. The rate of heat loss from the steam is Btu/s.arrow_forwardThe A/D converter wit the specifications listed below is planned to be used in an environment in which the A/D converter temperature may change by ± 10 °C. Estimate the contributions of conversion and quantization errors to the uncertainty in the digital representation of an analog voltage by the converter. FSO N Linearity error Temperature drift error Analog to Digital (A/D) Converter 0-10 V 12 bits ± 3 bits 1 bit/5 °Carrow_forward6-13. A smooth tube in the form of a circle of radius r rotates in its vertical plane with a constant angular velocity w. The position of a particle of mass m that slides inside the tube is given by the relative coordinate p. Find the differential equation for . e О E g ω Figure P6-13arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY





