A 1.05 -m-long rod of negligible weight is supported at its...

A 1.05 -m-long rod of negligible weight is supported at its ends by wires $A$ and $B$ of equal length (Fig. $\mathbf{P} \mathbf{1 1 . 8 3}$ ). The cross-sectional area of $A$ is $2.00 \mathrm{~mm}^{2}$ and that of $B$ is $4.00 \mathrm{~mm}^{2}$. Young's modulus for wire $A$ is $1.80 \times 10^{11} \mathrm{~Pa} ;$ that for $B$ is $1.20 \times 10^{11} \mathrm{~Pa}$. At what point along the rod should a weight $w$ be suspended to produce (a) equal stresses in $A$ and $B$ and (b) equal strains in $A$ and $B ?$

A 1.05 -m-long rod of negligible weight is supported at its ends by wires $A$ and $B$ of equal length (Fig. $\mathbf{P} \mathbf{1 1 . 8 3}$ ). The cross-sectional area of $A$ is $2.00 \mathrm{~mm}^{2}$ and that of $B$ is $4.00 \mathrm{~mm}^{2}$. Young's modulus for wire $A$ is $1.80 \times 10^{11} \mathrm{~Pa} ;$ that
for $B$ is $1.20 \times 10^{11} \mathrm{~Pa}$. At what point along the rod should a weight $w$ be suspended to produce (a) equal stresses in $A$ and $B$ and (b) equal strains in $A$ and $B ?$

Key Concepts

Moment Equilibrium

Moment equilibrium involves ensuring that the sum of moments (or torques) about any pivot point in a system is zero. This condition is used to analyze and design systems where loads create rotational effects, such as when a weight is suspended on a rod, to determine the correct placement of loads so that the resulting internal forces in supports meet specific criteria.

Young’s Modulus

Young’s modulus (or the modulus of elasticity) is a fundamental material property that defines the relationship between stress and strain in the elastic region. It provides a measure of a material’s stiffness, indicating how much it will deform under a given load. This concept is essential when comparing how different materials respond to the same stress level.

Force Equilibrium

Force equilibrium is a basic principle in statics stating that the sum of all forces acting on a system must equal zero for the system to be in equilibrium. This concept is applied to determine the force distribution in support structures when additional loads are applied, ensuring that the system remains balanced.

Stress

Stress is the internal force per unit area experienced by a material when subject to an external load. It is computed by dividing the force acting on the material by the cross?sectional area over which the force is distributed. Understanding stress is crucial because it determines whether a material will behave elastically or fail under load.

Strain

Strain is the measure of deformation representing the displacement between particles in the material body relative to a reference length. It quantifies how much a material stretches or compresses under stress. Being a dimensionless quantity, strain allows engineers to compare the relative change in shape or size of different materials under load.

Transcript

0:00 Height.

00:01 Here in this given problem, first of all this is the weightless rod whose length is given as 1 .05 meter.

00:15 Two wires suspending this rod, wires having equal length.

00:25 This is the wire a, this is the wire b.

00:29 Area of cross -section of wire a, that is given as 2 .00 millimeter square or we can say this this is 2 .00 multiplied by 10 to the par minus 6 meter square and area of cross section of wire b this is 4 .00 millimeter square or we can say this is 4 .00 multiplied by 10 raised to power minus 6 meter square.

00:59 Young's modulus of elasticity for wire a, that is 1 .80 into 10 raised to the part 11, pascal and for wire b this is 1 .20 into 10 to power 11 pascal.

01:20 Now suppose the weight w is attached at a distance x from the left end, from left end of the rod.

01:58 So in the first part of the problem, suppose tension in wire a, that is t, a and t, and that in wire b is t b.

02:31 Then stress in wire a that will be given by restoring force created in wire a which will be equal to the tension in wire a.

02:45 So restoring force per unit area of cross -section and stress in wire b that will be given by tb by a b.

02:59 In the first part of the problem it is given that stress in wire a is equal to that in wire b so we can say t a by a a is equal to t b by a b now total force upward means tension in both of the wires t a plus t b that should be equal to weight suspended as the rod itself is weightless.

03:32 So only weight w will be acting downward.

03:35 So using that we can say t a is equal to w minus t b and using equilibrium of rotation of the rod about its left end here.

04:12 This weight suspended will be creating a torque about this point clockwise in clockwise direction and this tension tb it will be creating a torque in counterclockwise direction so both of the torques should be equal in magnitude and opposite in direction so that the rod may not rotate so we can say for t b this is the product of tension t b with the length of the rod 1 .05 is equal to weight w multiplied by x so we can say t b is equal to w x divided by 1 .05 and hence t a will be given by that was w minus t b means this is w this is this t a is equal to w minus t b means w minus w minus w x x by 1 .05 or we can say this is w bracket 1 minus x by 1 .05...

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