Strong forces are needed to stretch a solid object. The force needed depends on several factors.

For two identical rods, more force is required to produce a greater amount of stretch than a smaller amount.

To stretch two equally long rods by the same amount, more force is needed for the rod with the larger cross-sectional area.

For a given amount of stretch, more force is required for a shorter rod than for a longer rod, provided they have the same cross-sectional area.

If the amount of stretching is small compared to the original length of the object: F = Y(ΔL/L 0 )A F is the magnitude of the stretching force, A is the cross-sectional area of the rod, L is the increase in length, L 0 is the original length,

Y is a proportionality constant called Young’s modulus, its value depends on the nature of the material. Young’s modulus has units of force per unit area (N/m 2 ).

The magnitude of the force in F = Y(ΔL/L 0 )A is proportional to the fractional increase in length ΔL/L 0 ), rather than the absolute increase ΔL.

The magnitude of the force is also proportional to the cross-sectional area A, which can be any shape, not just circular.

Forces that cause stretching create tension in the material and are called tensile forces.

This formula can also be used when a force compresses a material.

Ex. 1 - In a circus act, a performer supports the combined weight (1640 N) of a number of colleagues. Each femur of this performer has a length of 0.55 m and an effective cross-sectional area of 7.7 x m 2. Determine the amount by which each thighbone compresses under the extra weight.

Shear deformation occurs when the top of an object is pushed such that it shifts relative to the bottom of the object. This deformation is the combined effect of the top force +F and the force on the bottom of the object -F.

The formula is: F = S(ΔX/L 0 )A S is the shear modulus ΔX is the amount of shear L 0 is the thickness A is the area F is the force applied

Ex. 2 - A block of Jello is resting on a plate. It is m wide and long, and m thick. You push tangentially with a force of F= 0.45 N. The top surface moves ΔX = 6.0 x m relative to the bottom surface. Measure the shear modulus of Jello.

Young’s modulus refers to a change in length of one dimension of a solid object while the shear modulus refers to a change in shape of a solid object as a result of shearing forces.

Pressure can be applied to all surfaces of an object and compress it in all dimensions at once. Pressure P is the perpendicular force per unit area.

Pressure is the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts: P = F/A Unit: N/m 2 = pascal (Pa)

A pressure increase ΔP decreases the volume of an object by ΔV. ΔP = -B(ΔV/V 0 ) B is the bulk modulus.

Ex. 2 - The water pressure at the bottom of the Mariana trench is ΔP = 1.1 x 10 8 Pa greater than the pressure at the surface. A solid steel ball of volume V 0 = 0.20 m 3 sinks to the bottom of the trench. What is the change ΔV when it reaches the bottom?

F/A = Y(ΔL/L 0 ) F/A = S(ΔX/L 0 ) ΔP = -B(ΔV/V 0 )

F/A = Y(ΔL/L 0 ) F/A = S(ΔX/L 0 ) ΔP = -B(ΔV/V 0 ) The ratio of force to area is called stress. The delta ratios in the above parentheses are called strain. If the moduli are kept constant, these equations apply to a wide range of materials. Therefore, stress and strain are directly proportional to one another.

This relationship is called Hooke’s Law. Stress is directly proportional to strain. Unit of stress: newton per square meter = pascal (Pa). Strain is unit less.

As long as stress remains proportional to strain, a plot of stress vs. strain is a straight line. Actually there is a point at which stress vs. strain are no longer proportional, this is called the “proportionality limit”.

Once the stress is removed, the object will return to its original shape. If the “elastic limit” is exceeded, the object will not return to its original shape and will be permanently deformed.

Springs exhibit elastic behavior. For most springs: F = kx x is how much the spring is displaced from its original length, k is the spring constant (N/m), F is the force.

From F/A = Y(ΔL/L 0 ) and F = kx: F = YA/L 0 ΔL ΔL is x, and YA/L 0 is k

A spring that behaves according to Hooke’s Law is called an ideal spring.

Ex. 4 - In a tire pressure gauge, the air in the tire pushes against a plunger attached to a spring when the gauge is pressed against the tire valve. Suppose the spring constant of the of the spring is k = 320 N/m and the bar indicator of the gauge extends 2.0 cm when the gauge is pressed against the air valve. What force does the air in the tire apply to the spring?

The spring constant k is often referred to as the “stiffness” of the spring.

A force must be applied to a spring to stretch or compress it. By Newton’s third law, the spring must apply an equal force to whatever is applying the force to the spring.

This reaction force is often called the “restoring force” and is represented by the equation F = -kx.

This force varies with the displacement. Therefore the acceleration varies with the displacement.

When the restoring force has the mathematical form given by F = -kx, the type of motion resulting is called “simple harmonic motion”.

A graph of this motion is sinusoidal. When an object is hung from a spring, the equilibrium position is determined by how far the weight stretches the spring initially.

F = kd 0 becomes mg = kd 0 or d 0 = mg/k