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Elasticity and Strength of Materials. The effect of forces on the shape of the body. When a force is applied to a body, the shape and size of the body change. Depending on how the force is applied, the body may be stretched , compressed bent or twisted.
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When a force is applied to a body, the shape and size of the body change. Depending on how the force is applied, the body may be stretched, compressedbent ortwisted. Elasticity is the property of a body that tends to return the body to its original shape after the force is removed. If the applied force is sufficiently large, however, the body is distorted beyond its elastic limit, and the original shape is not restored after removal of the force. A still larger force will rupture the body
Longitudinal Stretch and Compression Let us consider the effect of a stretching force F applied to a bar. The applied force is transmitted to every part of the body, and it tends to pull the material apart. This force, however, is resisted by the cohesive force that holds the material together. The material breaks when the applied force exceeds the cohesive force. If the force is reversed, the bar is compressed, and its length is reduced. A sufficiently large force will produce permanent deformation and then breakage.
Longitudinal Stretch and Compression Robert Hooke Young’s modulus Stress S is defined as The force applied to the bar causes the bar to elongate by an amount Δl. The fractional change in length Δl/l is called the longitudinal strainSt; that is In 1676 Robert Hooke observed that while the body remains elastic, the ratio of stress to strain is constant (Hooke’s law); that is,
A Spring The force F required to stretch (or compress) the spring is directly proportional to the amount of stretch; that is K: the spring constant A stretched (or compressed) spring contains potential energy. The energy E stored in the spring is given by
An elastic body under stress is analogous to a spring with a spring constant YA/l. By analogy with the spring, the amount of energy stored in a stretched or compressed body is
Bone Fracture: Energy Considerations The corresponding force FB that will fracture the bone is, Knowledge of the maximum energy that parts of the body can safely absorb allow us to estimate the possibility of injury under various circumstances. We shall first calculate the amount of energy required to break a bone of area A and lengthl. Assume that the bone remains elastic until fracture. The compression Δl at the breaking point is, therefore
Bone Fracture: Energy Considerations • 1. A = 6 cm2 • 2. SB= 109 dyn/cm2 • 3.Y = 14 ×1010 dyn/cm2 Data used Consider the fracture of two leg bones that have a combined length of about 90 cm and an average area of about 6 cm2. The combined energy in the two legs is twice this value, or 385 J. This is the amount of energy in the impact of a 70-kg person jumping from a height 0f 56 cm. By bending the joints of the body we can jump from a height larger than 56 cm and without any injury.
Impulsive Forces In a sudden collision, a large force is exerted for a short period of time on the colliding object. The force starts at zero, increases to some maximum value, and then decreases to zero again in a very short period of time.
Impulsive Forces Duration of the collision Such a short-duration force is called an impulsive force. The average value of the impulsive force Favcan be calculated. For example, if the duration of a collision is 6 × 10-3 sec and the change in momentum is 2 kg m/sec, the average force that acted during the collision is
Fracture Due to a Fall: Impulsive Force Considerations Calculation of injured effect using the concept of impulsive force When a person falls from a height, his/her velocity on impact with the ground, neglecting air friction, is The momentum on impact is After the impact, the change in momentum is The average impact force is
Fracture Due to a Fall: Impulsive Force Considerations For a man of 70 kg If the impact surface is hard, such as concrete, and if the person falls with his/her joints rigidly locked, the collision time is estimated to be about 10-2 sec. The breaking stress that may cause a bone fracture is 109 dyne/cm2. If the person falls flat on his/her heels, the area of impact may be about 2cm2. The force FB that will cause fracture is
Car Accident Lamborghini
Airbags Air Bag
Airbags: Inflating Collision Protection Devices The average deceleration is given by The average force At an impact velocity of 70 km/h, F = 4.45 × 109 dyn. If this force is uniformly distributed over a 1000-cm2 area of the passenger’s body, the stress, S, is 4.45 × 106 dyn/cm2. This is just below the estimated strength of body tissue. An inflatable bag is located in the dashboard of the car. In a collision, the bag expands, suddenly and cushions the impact of the passenger. The forward motion of the passenger must be stopped in about 30 cm of motion if contact with the hard surfaces of the car is to be avoided. where v is the initial velocity of the automobile and s is the distance over which the deceleration occurs.
Airbags: Inflating Collision Protection Devices If v = 105 km/h stress Such a force would probably injure the passenger.
Whiplash Injury Neck bones are rather delicate and can be fractured by even a moderate force. Fortunately the neck muscles are relatively strong and are strong and are capable of absorbing a considerable amount of energy. If, however, the impact is sudden, the body is accelerated in the forward direction by the back of the seat, and the unsupported neck is then suddenly yanked back at full speed. Here the muscles do not respond fast enough and all the energy is absorbed by the neck bones, causing the well-known whiplash injury. Exercise 5-5
Insect Wing Muscles Wing Joint The wing movement is controlled by many muscles, which are here represented by muscles A and B. The upward movement of the wings is produced by the contraction of muscle A, which depresses the upper part of the thorax and causes the attached wings to move up. While muscle A contracts, muscle B is relaxed. Note that the force produced by the muscle A is applied to the wing by means of a Class 1 lever. The fulcrum here is the wing joint marked by the small circle in the figure. A number of different wing-muscle arrangements occur in insects. One of a highly simplified arrangement is found in the dragonfly. Upward Movement
Insect Wing Muscles Wing Joint This is made possible by the lever arrangement of the wings. Downward Movement The downward wing movement is produced by the contraction of muscle B while muscle A is relaxed. Here the force is applied to the wings by means of a Class 3 lever. The physical characteristics of insect flight muscles are not peculiar to insects. The amount of force per unit area of the muscle and the rate of muscle contraction are similar to the values measured for human muscles. Yet insect wing muscles are required to flap the wings at a very high rate.
Hovering Flight The wings of most insects are designed so that during the upward stroke the force on the wings is small. During the upward movement of the wings, the gravitational force causes the insect to drop. The downward wing movement then produces an upward force that restores the insect to its original position. The vertical position of the insect thus oscillates up and down at the frequency of the wing-beat. The distance the insect falls between wing-beats depends on how rapidly its wings are beating. If the insect flaps its wings at a slow rate, the time interval during which the lifting force is zero is longer, and therefore the insect falls farther than its wings were beating rapidly.
Hovering Flight We want to compute the wing-beat frequency necessary for the insect to maintain a given stability in its amplitude. Assuming that the lifting force is at a finite constant value while the wings are moving down and that it is zero while the wings are moving up. During the time interval Δtof the upward wing-beat , the insect drops a distance h under the action of gravity. The downward stroke then restores the insect to its original position. Typically, it may be required that the vertical position of the insect change by no more than 0.1 mm (i.e. h = 0.1 mm).
This is a typical insect wing-beat frequency. Since the up movements and the down movements of the wings are about equal in duration, the period T for a complete up-and-down wing movement is twice Δt, that is, The frequency of wing-beats f, is Frequency of Bee Wings
Elasticity of Wings As the wings are accelerated, they gain kinetic energy, which is provided by the muscles. When the wings are decelerated toward the end of the stroke, this energy must be dissipated. During the down stroke, the kinetic energy is dissipated by the muscles and is converted into heat. Some insects are able to utilize the kinetic energy in the upward movement of the wings to aid in their flight and this has to do with a kind of rubberlike protein called resilin.
