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Learning Goal: To calculate the angle of twist at multiple points in a stepped solid circular shaft. For a shaft of length fixed at one end and subject to torque around its axis, the L T(z) dz angle of twist at the free end of the shaft is = √o (z)G(z) where T(I) is the internal torque, J(z) is the polar moment of inertia, and G(z) is the shear modulus of the material and is the distance from the fixed end. This equation applies as long as the response is linear elastic and the cross section does not change too suddenly. In the simpler case of a constant cross section, single material, and constant internal TL JG torque, the integral can be evaluated to give = T. This shows that the angle of twist is linear with respect to the internal torque and the length of the shaft. If a shaft has multiple segments where the torque, cross section, and properties are constant, then the total angle of twist at the free end can be written as a sum of the angles of twist for each part, =ΣT The sign convention for the internal torque and angles of twist follows the right-hand rule (Figure 1). A torque or twist is considered positive when the thumb of a right hand is directed outward from the section while the fingers curl in the direction of the torque or twist. The circular shaft shown (Figure 2) has dimensions d₁ = 24 cm. L₁ = 6 m, d₂ = 18.5 cm, and L2 = 4 m with applied torques T₁ = 100 kN·m and T₂ = 40 kN.m. The shear modulus is G = 60 GPa Point B is halfway between points A and C. ▸ Part A Part B - Angle of twist at B What is the angle of twist at point B? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) НА = 0.0525 rad ? Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining Part C - Angle of twist at D For the given shaft, what is the angle of twist at the free end, point D? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) фр= Value Units Submit ?

Learning Goal: To calculate the angle of twist at multiple points in a stepped solid circular shaft. For a shaft of length fixed at one end and subject to torque around its axis, the L T(z) dz angle of twist at the free end of the shaft is = √o (z)G(z) where T(I) is the internal torque, J(z) is the polar moment of inertia, and G(z) is the shear modulus of the material and is the distance from the fixed end. This equation applies as long as the response is linear elastic and the cross section does not change too suddenly. In the simpler case of a constant cross section, single material, and constant internal TL JG torque, the integral can be evaluated to give = T. This shows that the angle of twist is linear with respect to the internal torque and the length of the shaft. If a shaft has multiple segments where the torque, cross section, and properties are constant, then the total angle of twist at the free end can be written as a sum of the angles of twist for each part, =ΣT The sign convention for the internal torque and angles of twist follows the right-hand rule (Figure 1). A torque or twist is considered positive when the thumb of a right hand is directed outward from the section while the fingers curl in the direction of the torque or twist. The circular shaft shown (Figure 2) has dimensions d₁ = 24 cm. L₁ = 6 m, d₂ = 18.5 cm, and L2 = 4 m with applied torques T₁ = 100 kN·m and T₂ = 40 kN.m. The shear modulus is G = 60 GPa Point B is halfway between points A and C. ▸ Part A Part B - Angle of twist at B What is the angle of twist at point B? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) НА = 0.0525 rad ? Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining Part C - Angle of twist at D For the given shaft, what is the angle of twist at the free end, point D? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) фр= Value Units Submit ?

University Physics Volume 1
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter10: Fixed-axis Rotation
Section: Chapter Questions
Problem 7CQ: If the angular acceleration of a rigid body is zero, what is the functional form of the angular...
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Learning Goal: To calculate the angle of twist at multiple points in a stepped solid circular shaft. For a shaft of length fixed at one end and subject to torque around its axis, the L T(z) dz angle of twist at the free end of the shaft is = √o (z)G(z) where T(I) is the internal torque, J(z) is the polar moment of inertia, and G(z) is the shear modulus of the material and is the distance from the fixed end. This equation applies as long as the response is linear elastic and the cross section does not change too suddenly. In the simpler case of a constant cross section, single material, and constant internal TL JG torque, the integral can be evaluated to give = T. This shows that the angle of twist is linear with respect to the internal torque and the length of the shaft. If a shaft has multiple segments where the torque, cross section, and properties are constant, then the total angle of twist at the free end can be written as a sum of the angles of twist for each part, =ΣT The sign convention for the internal torque and angles of twist follows the right-hand rule (Figure 1). A torque or twist is considered positive when the thumb of a right hand is directed outward from the section while the fingers curl in the direction of the torque or twist. The circular shaft shown (Figure 2) has dimensions d₁ = 24 cm. L₁ = 6 m, d₂ = 18.5 cm, and L2 = 4 m with applied torques T₁ = 100 kN·m and T₂ = 40 kN.m. The shear modulus is G = 60 GPa Point B is halfway between points A and C. ▸ Part A Part B - Angle of twist at B What is the angle of twist at point B? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) НА = 0.0525 rad ? Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining Part C - Angle of twist at D For the given shaft, what is the angle of twist at the free end, point D? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) фр= Value Units Submit ?
Transcribed Image Text:Learning Goal: To calculate the angle of twist at multiple points in a stepped solid circular shaft. For a shaft of length fixed at one end and subject to torque around its axis, the L T(z) dz angle of twist at the free end of the shaft is = √o (z)G(z) where T(I) is the internal torque, J(z) is the polar moment of inertia, and G(z) is the shear modulus of the material and is the distance from the fixed end. This equation applies as long as the response is linear elastic and the cross section does not change too suddenly. In the simpler case of a constant cross section, single material, and constant internal TL JG torque, the integral can be evaluated to give = T. This shows that the angle of twist is linear with respect to the internal torque and the length of the shaft. If a shaft has multiple segments where the torque, cross section, and properties are constant, then the total angle of twist at the free end can be written as a sum of the angles of twist for each part, =ΣT The sign convention for the internal torque and angles of twist follows the right-hand rule (Figure 1). A torque or twist is considered positive when the thumb of a right hand is directed outward from the section while the fingers curl in the direction of the torque or twist. The circular shaft shown (Figure 2) has dimensions d₁ = 24 cm. L₁ = 6 m, d₂ = 18.5 cm, and L2 = 4 m with applied torques T₁ = 100 kN·m and T₂ = 40 kN.m. The shear modulus is G = 60 GPa Point B is halfway between points A and C. ▸ Part A Part B - Angle of twist at B What is the angle of twist at point B? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) НА = 0.0525 rad ? Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining Part C - Angle of twist at D For the given shaft, what is the angle of twist at the free end, point D? Express your answer with appropriate units to three significant figures. ▸ View Available Hint(s) фр= Value Units Submit ?
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