Q: A 70-kg person in a car crash is stopped by an airbag over a distance of 30 cm. If the impact velocity is $$v$$, what is the average stopping force $$F$$?

$$F = \frac{70 \times 10^3$ v^2$}{2 \times 30}$$

Q: If the Young's modulus of bone is $$14 \times 10$$^{10}$$ \text{dyn}/\text{cm}^2$$ and the rupture strength is $$10^6$$ \text{dyn}/\text{cm}^2$$, what does this imply about bone's ability to withstand stress?

Bone will fracture when the stress exceeds $$10^6$$ \text{dyn}/\text{cm}^2$$.

Question 1

A steel rod of length $$l$$ and cross-sectional area $$A is $$subjected to a tensile force $$F. $$Which equation correctly expresses the longitudinal strain $$S_l in $$the rod?

Accepted Answer

$$S_l = \frac{\Delta l}{l}$$

Question 2

Which of the following best describes Young's modulus $$Y$$ for a material?

Accepted Answer

$$Y = \frac{S}{S_l}$$

Question 3

A spring with spring constant $$K is $$stretched by a distance $$\Delta l. $$What is the potential energy stored in the spring?

Accepted Answer

$$E = \frac{1}{2} K \Delta l^2$

Question 4

If a bone of length $$l$$ and area $$A$$ fractures at a breaking stress $$S_B$$, what is the force $$F_B$$ required to fracture the bone?

Accepted Answer

$$F_B = S_B A$$

Question 5

A person falls from a height $$h. $$What is their velocity $$v$$ upon impact with the ground, neglecting air resistance?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 6

Which equation gives the average impulsive force $$F_{av}$$ exerted on an object during a collision, given initial and final velocities $$v_i$$ and $$v_f$$ over time $$\Delta t$$?

Accepted Answer

$$F_{av} = \frac{m(v_f - v_i)}{\Delta t}$$

Question 7

A 70-kg person falls from a height of 56 cm. What is the energy absorbed by the bones in both legs at the point of compressive fracture, given $$S_B = 10$$^{6}$$ 	ext{dyn}/	ext{cm}^2$$, $$Y = 14 	imes 10$$^{10}$$ 	ext{dyn}/	ext{cm}^2$$, $$A = 6 	ext{cm}^2$$, and $$l = 90 	ext{cm}$$?

Accepted Answer

$$385 	ext{J}$$

Question 8

Which formula correctly expresses the compression $$\Delta l at $$the breaking point of a bone?

Accepted Answer

$$\Delta l = \frac{S_B l}{Y}$$

Question 9

In a car collision, the airbag expands to stop the passenger's forward motion in a distance $$s. If $$the passenger's velocity is $$v$$, what is the average deceleration $$a$$?

Accepted Answer

$$a = \frac{v^2$}{2s}$$

Question 10

A stretched elastic body is analogous to a spring. What is the effective spring constant $$K$$ for a rod of Young's modulus $$Y$$, area $$A$$, and length $$l$$?

Accepted Answer

$$K = \frac{Y A}{l}$$

Question 11

Which equation gives the energy stored in a compressed bone at the point of fracture?

Accepted Answer

$$E = \frac{1}{2} \frac{A l S_B^2$}{Y}$$

Question 12

A 70-kg person in a car crash is stopped by an airbag over a distance of 30 cm. If the impact velocity is $$v$$, what is the average stopping force $$F$$?

Accepted Answer

$$F = \frac{70 	imes 10^3$ v^2$}{2 	imes 30}$$

Question 13

Which of the following statements about Hooke's law is correct?

Accepted Answer

The ratio of stress to strain is constant only within the elastic limit.

Question 14

If the average stopping force in a car crash is distributed over a $$1000 	ext{cm}^2$$ area of the passenger's body, and the force per unit area is $$4.42 	imes 10^6$ 	ext{dyn}/	ext{cm}^2$$, what does this value represent?

Accepted Answer

It is just below the estimated strength of body tissue.

Question 15

A spring is stretched by a force $$F. $$Which equation correctly relates the force to the spring constant $$K$$ and the extension $$\Delta l$$?

Accepted Answer

$$F = -K \Delta l$$

Question 16

A person falls from a height $$h$$ and comes to rest in time $$\Delta t. $$What is the average impact force $$F on $$the ground?

Accepted Answer

$$F = \frac{m}{\Delta t} \sqrt{2gh}$$

Question 17

Which of the following is a correct real-world application of the concept of impulsive force?

Accepted Answer

Estimating the force during a car crash when the collision time is very short.

Question 18

If the Young's modulus of bone is $$14 	imes 10$$^{10}$$ 	ext{dyn}/	ext{cm}^2$$ and the rupture strength is $$10^6$$ 	ext{dyn}/	ext{cm}^2$$, what does this imply about bone's ability to withstand stress?

Accepted Answer

Bone will fracture when the stress exceeds $$10^6$$ 	ext{dyn}/	ext{cm}^2$$.

Elasticity and Strength of Materials: Study Notes

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