Unit 1 Engineering Physics

36 CHAPTER 1. PROPERTIES OF MATTER

l

Load

Pin

Figure 1.32: Non-uniform bending - experimental set up

distance between the two knife edges and let a weightW = Mgis suspended at the
midpoint (i.e., at the distancel/ 2 from the knife edges). Then the reaction at each knife
edge isW/ 2 which acts vertically upwards. The part of the beam between the knife edges
is equivalent to a pair of inverted cantilevers, each of lengthl/ 2 with an upward force
W/ 2 at the inverted free end as shown in Figure1.33. The elevation at the loaded end

l

Load

l 2 <latexit sha1_base64="b8OymSCgBm9xer8Rj4ow1i6PWmA=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJbzQFofNWa1gVv16l4BtE78Jak2b6BAa+B+BkNJsoQKQzjWuu97qQlzrAwjnM4qQaZpiskEj2jfUoETqsO8uHmGLqwyRLFUtoRBhfp7IseJ1tMksp0JNmO96s3F/7x+ZuLrMGcizQwVZLEozjgyEs0DQEOmKDF8agkmitlbERljhYmxMVVsCP7qy+uk06j7Xt2/b/ykAWU4g3O4BB+uoAl30II2EEjhCV7g1cmcZ+fNeV+0lpzlzCn8gfPxDcAOkYg=</latexit><latexit sha1_base64="b8OymSCgBm9xer8Rj4ow1i6PWmA=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJbzQFofNWa1gVv16l4BtE78Jak2b6BAa+B+BkNJsoQKQzjWuu97qQlzrAwjnM4qQaZpiskEj2jfUoETqsO8uHmGLqwyRLFUtoRBhfp7IseJ1tMksp0JNmO96s3F/7x+ZuLrMGcizQwVZLEozjgyEs0DQEOmKDF8agkmitlbERljhYmxMVVsCP7qy+uk06j7Xt2/b/ykAWU4g3O4BB+uoAl30II2EEjhCV7g1cmcZ+fNeV+0lpzlzCn8gfPxDcAOkYg=</latexit><latexit sha1_base64="b8OymSCgBm9xer8Rj4ow1i6PWmA=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJbzQFofNWa1gVv16l4BtE78Jak2b6BAa+B+BkNJsoQKQzjWuu97qQlzrAwjnM4qQaZpiskEj2jfUoETqsO8uHmGLqwyRLFUtoRBhfp7IseJ1tMksp0JNmO96s3F/7x+ZuLrMGcizQwVZLEozjgyEs0DQEOmKDF8agkmitlbERljhYmxMVVsCP7qy+uk06j7Xt2/b/ykAWU4g3O4BB+uoAl30II2EEjhCV7g1cmcZ+fNeV+0lpzlzCn8gfPxDcAOkYg=</latexit><latexit sha1_base64="wokpPsFO4eJyF0c4Bt8tHEf2sUw=">AAAB83icbVBNTwIxEJ3iF+IX6tFLI5h4Irtc9Ej04hETARN2Q7qlCw3ddtN2TciGv+HFg8Z49c94899YYA8KvmSSl/dmMjMvSgU31vO+UWljc2t7p7xb2ds/ODyqHp90jco0ZR2qhNKPETFMcMk6llvBHlPNSBIJ1osmt3O/98S04Uo+2GnKwoSMJI85JdZJQT0XgXI+bs7qg2rNa3gL4HXiF6QGBdqD6lcwVDRLmLRUEGP6vpfaMCfacirYrBJkhqWETsiI9R2VJGEmzBc3z/CFU4Y4VtqVtHih/p7ISWLMNIlcZ0Ls2Kx6c/E/r5/Z+DrMuUwzyyRdLoozga3C8wDwkGtGrZg6Qqjm7lZMx0QTal1MFReCv/ryOuk2G77X8O+btdZNEUcZzuAcLsGHK2jBHbShAxRSeIZXeEMZekHv6GPZWkLFzCn8Afr8AQ+YkQY=</latexit>

W 2 <latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="x6+ivFJsBtQwU+2O0yzfvkMv2Ys=">AAAB83icbVBNTwIxEJ3iF+IX6tFLI5h4Irtc9Ej04hETARN2Q7qlCw3ddtN2TciGv+HFg8Z49c94899YYA8KvmSSl/dmMjMvSgU31vO+UWljc2t7p7xb2ds/ODyqHp90jco0ZR2qhNKPETFMcMk6llvBHlPNSBIJ1osmt3O/98S04Uo+2GnKwoSMJI85JdZJQT3vBcr5uDmrD6o1r+EtgNeJX5AaFGgPql/BUNEsYdJSQYzp+15qw5xoy6lgs0qQGZYSOiEj1ndUkoSZMF/cPMMXThniWGlX0uKF+nsiJ4kx0yRynQmxY7PqzcX/vH5m4+sw5zLNLJN0uSjOBLYKzwPAQ64ZtWLqCKGau1sxHRNNqHUxVVwI/urL66TbbPhew79v1lo3RRxlOINzuAQfrqAFd9CGDlBI4Rle4Q1l6AW9o49lawkVM6fwB+jzB+74kPE=</latexit>

W 2 <latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="HzL8q1sPfmxum/MGxTNppSgFBJ4=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LLaCp5L0oseiF48V7Ac0oWy2m3bpZjfsboQS+je8eFDEq3/Gm//GbVpEWx8MPN6bYWZelHKmjed9OaWNza3tnfJuZW//4PDIPT7paJkpQttEcql6EdaUM0HbhhlOe6miOIk47UaT27nffaRKMykezDSlYYJHgsWMYGOloJZ3A2l91JjVBm7Vq3sF0Drxl6TavIECrYH7GQwlyRIqDOFY677vpSbMsTKMcDqrBJmmKSYTPKJ9SwVOqA7z4uYZurDKEMVS2RIGFerviRwnWk+TyHYm2Iz1qjcX//P6mYmvw5yJNDNUkMWiOOPISDQPAA2ZosTwqSWYKGZvRWSMFSbGxlSxIfirL6+TTqPue3X/vvGTBpThDM7hEny4gibcQQvaQCCFJ3iBVydznp03533RWnKWM6fwB87HN599kXM=</latexit><latexit sha1_base64="x6+ivFJsBtQwU+2O0yzfvkMv2Ys=">AAAB83icbVBNTwIxEJ3iF+IX6tFLI5h4Irtc9Ej04hETARN2Q7qlCw3ddtN2TciGv+HFg8Z49c94899YYA8KvmSSl/dmMjMvSgU31vO+UWljc2t7p7xb2ds/ODyqHp90jco0ZR2qhNKPETFMcMk6llvBHlPNSBIJ1osmt3O/98S04Uo+2GnKwoSMJI85JdZJQT3vBcr5uDmrD6o1r+EtgNeJX5AaFGgPql/BUNEsYdJSQYzp+15qw5xoy6lgs0qQGZYSOiEj1ndUkoSZMF/cPMMXThniWGlX0uKF+nsiJ4kx0yRynQmxY7PqzcX/vH5m4+sw5zLNLJN0uSjOBLYKzwPAQ64ZtWLqCKGau1sxHRNNqHUxVVwI/urL66TbbPhew79v1lo3RRxlOINzuAQfrqAFd9CGDlBI4Rle4Q1l6AW9o49lawkVM6fwB+jzB+74kPE=</latexit> y

W

Figure 1.33: Non-uniform bending - Equivalence to a pair of inverted cantilevers

of each inverted cantilever is equal to the depressionyat the midpoint of the bar. The
Young’s ModulusY is can be written from equation (1.33)as,

Y =

(^1) W
2
(^21) l
2
(^23)
3 yIg

(1.41)

For a beam of rectangular cross-section with breadthband thicknessd,

Y =

Mgl^3 48 y

(^1) bd 3
12
2
Therefore, the Young’s Modulus of a nonuniformly bent bar is given by,

Y=

Mgl^3 4 bd^3 y

(1.42)

Experimental Procedure

As shown in Figure1.32, support the given beam symmetrically on two knife edges with

36

Unit 1 Engineering Physics

36 CHAPTER 1. PROPERTIES OF MATTER

Y =

(1.41)

Y =

Y=

(1.42)

Experimental Procedure

36

Get our desktop app

Company

Features

Documentation

Resources