Section 5.5
Section 5.5
Fracture Due to a Fall: Impulsive Force Considerations and the change in momentum is 2 kg m/sec, the average force that acted during the collision is Fav 2 kg m/sec 3.3 × 102 N 6 × 10−3 sec
Note that, for a given momentum change, the magnitude of the impulsive force is inversely proportional to the collision time; that is, the collision force is larger in a fast collision than in a slower collision.
5.5
Fracture Due to a Fall: Impulsive Force
Considerations
In the preceding section, we calculated the injurious effects of collisions from energy considerations. Similar calculations can be performed using the concept of impulsive force. The magnitude of the force that causes the damage is computed from Eq. 5.14. The change in momentum due to the collision is usually easy to calculate, but the duration of the collision t is difficult to determine precisely. It depends on the type of collision. If the colliding objects are hard, the collision time is very short, a few milliseconds. If one of the objects is soft and yields during the collision, the duration of the collision is lengthened, and as a result the impulsive force is reduced. Thus, falling into soft sand is less damaging than falling on a hard concrete surface.
When a person falls from a height h, his/her velocity on impact with the ground, neglecting air friction (see Eq. 3.6), is
v
2gh
(5.15)
The momentum on impact is
2h
mv m 2gh W
(5.16)
g
After the impact the body is at rest, and its momentum is therefore zero (mvf 0). The change in momentum is
2h
mvi − mvf W
(5.17)
g
The average impact force, from Eq. 5.14, is
2h
F W
m 2gh
(5.18)
t
g
t
Chapter 5
Elasticity and Strength of Materials
Now comes the difficult part of the problem: Estimate of the collision duration. 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 collision time is considerably longer if the person bends his/her knees or falls on a soft surface.
From Table 5.1, the force per unit area that may cause a bone fracture is 109 dyn/cm2. If the person falls flat on his/her heels, the area of impact may be about 2 cm2. Therefore, the force FB that will cause fracture is FB 2 cm2 × 109 dyn/cm2 2 × 109 dyn (4.3 × 103 lb) From Eq. 5.18, the height h of fall that will produce such an impulsive force is given by
2
Fth 1
(5.19)
2g
m
For a man with a mass of 70 kg, the height of the jump that will generate a fracturing average impact force (assuming t 10−2 sec) is given by
2
Ft
2 × 109 × 10−2 h 1
1
41.6 cm (1.37 ft)
2g
m
2 × 980 70 × 103
This is close to the result that we obtained from energy considerations. Note, however, that the assumption of a 2-cm2 impact area is reasonable but somewhat arbitrary. The area may be smaller or larger depending on the nature of the landing; furthermore, we have assumed that the person lands with legs rigidly straight. Exercises 5-2 and 5-3 provide further examples of calculating the injurious effect of impulsive forces.
5.6Airbags: Inflating Collision Protection Devices
The impact force may also be calculated from the distance the center of mass of the body travels during the collision under the action of the impulsive force. This is illustrated by examining the inflatable safety device used in automobiles (see Fig. 5.5). 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.
The average deceleration (see Eq. 3.6) is given by a v2
(5.20)
2s
Fracture Due to a Fall: Impulsive Force Considerations and the change in momentum is 2 kg m/sec, the average force that acted during the collision is Fav 2 kg m/sec 3.3 × 102 N 6 × 10−3 sec
Note that, for a given momentum change, the magnitude of the impulsive force is inversely proportional to the collision time; that is, the collision force is larger in a fast collision than in a slower collision.
5.5
Fracture Due to a Fall: Impulsive Force
Considerations
In the preceding section, we calculated the injurious effects of collisions from energy considerations. Similar calculations can be performed using the concept of impulsive force. The magnitude of the force that causes the damage is computed from Eq. 5.14. The change in momentum due to the collision is usually easy to calculate, but the duration of the collision t is difficult to determine precisely. It depends on the type of collision. If the colliding objects are hard, the collision time is very short, a few milliseconds. If one of the objects is soft and yields during the collision, the duration of the collision is lengthened, and as a result the impulsive force is reduced. Thus, falling into soft sand is less damaging than falling on a hard concrete surface.
When a person falls from a height h, his/her velocity on impact with the ground, neglecting air friction (see Eq. 3.6), is
v
2gh
(5.15)
The momentum on impact is
2h
mv m 2gh W
(5.16)
g
After the impact the body is at rest, and its momentum is therefore zero (mvf 0). The change in momentum is
2h
mvi − mvf W
(5.17)
g
The average impact force, from Eq. 5.14, is
2h
F W
m 2gh
(5.18)
t
g
t
Chapter 5
Elasticity and Strength of Materials
Now comes the difficult part of the problem: Estimate of the collision duration. 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 collision time is considerably longer if the person bends his/her knees or falls on a soft surface.
From Table 5.1, the force per unit area that may cause a bone fracture is 109 dyn/cm2. If the person falls flat on his/her heels, the area of impact may be about 2 cm2. Therefore, the force FB that will cause fracture is FB 2 cm2 × 109 dyn/cm2 2 × 109 dyn (4.3 × 103 lb) From Eq. 5.18, the height h of fall that will produce such an impulsive force is given by
2
Fth 1
(5.19)
2g
m
For a man with a mass of 70 kg, the height of the jump that will generate a fracturing average impact force (assuming t 10−2 sec) is given by
2
Ft
2 × 109 × 10−2 h 1
1
41.6 cm (1.37 ft)
2g
m
2 × 980 70 × 103
This is close to the result that we obtained from energy considerations. Note, however, that the assumption of a 2-cm2 impact area is reasonable but somewhat arbitrary. The area may be smaller or larger depending on the nature of the landing; furthermore, we have assumed that the person lands with legs rigidly straight. Exercises 5-2 and 5-3 provide further examples of calculating the injurious effect of impulsive forces.
5.6Airbags: Inflating Collision Protection Devices
The impact force may also be calculated from the distance the center of mass of the body travels during the collision under the action of the impulsive force. This is illustrated by examining the inflatable safety device used in automobiles (see Fig. 5.5). 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.
The average deceleration (see Eq. 3.6) is given by a v2
(5.20)
2s