In the Grimms’ fairy tale Rapunzel, she lets down her golden hair to a length of $20$ yards (we’ll use $20 m$, which is not much different) so that the prince can climb up to her room. Human hair has a Young's modulus of about $490 \mathrm{MPa}$, and we can assume that Rapunzel's hair can be squeezed into a rope about $2.0 \mathrm{~cm}$ in cross-sectional diameter. The prince is described as young and handsome, so we can estimate a mass of $60 \mathrm{~kg}$ for him.
(a) Just after the prince has started to climb at a constant speed, while he is still near the bottom of the hair, by how many centimeters does he stretch Rapunzel's hair?
(b) What is the mass of the heaviest prince that could climb up, given that the maximum tensile stress hair can support is $196 \mathrm{MPa}$?

Tendon-Stretching Exercises. As part of an exercise program, a $75-\mathrm{kg}$ person does toe raises in which he raises his entire body weight on the ball of one foot . The Achilles tendon pulls straight upward on the heel bone of his foot. This tendon is $25 \mathrm{~cm}$ long and has a cross-sectional area of $78 \mathrm{~mm}^2$ and a Young's modulus of $1470 \mathrm{MPa}$. What force does the Achilles tendon exert on the heel during this exercise? Express your answer in newtons and in multiples of his weight.

As part of an exercise program, a $75-\mathrm{kg}$ person does toe raises in which he raises his entire body weight on the ball of one foot (Fig. P11.59). The Achilles tendon pulls straight upward on the heel bone of his foot. This tendon is $25 \mathrm{~cm}$ long and has a cross-sectional area of $78 \mathrm{~mm}^2$ and a Young's modulus of $1470 \mathrm{MPa}$. Draw a free-body diagram of the person's foot (everything below the ankle joint). Ignore the weight of the foot.

You hang a floodlamp from the end of a vertical steel wire. The floodlamp stretches the wire $0.18 \mathrm{~mm}$ and the stress is proportional to the strain. How much would it have stretched $(a)$ if the wire were twice as long? $(b)$ if the wire had the same length but twice the diameter? $(c)$ for a copper wire of the original length and diameter?

The compres- sive strength of our bones is important in everyday life. Young's modulus for bone is about $1.4 \times 10^{10} \mathrm{~Pa}$. Bone can take only about a $1.0\%$ change in its length before fracturing.
(b) Estimate the maximum height from which a $70$-$\mathrm{kg}$ man could jump and not fracture his tibia. Take the time between when he first touches the floor and when he has stopped to be $0.030$ s, and assume that the stress on his two legs is distributed equally.

The compressive strength of our bones is important in everyday life. Young’s modulus for bone is about $1.4 \times 10^{10} \mathrm{Pa}$. Bone can take only about a 1.0% change in its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is $3.0 \mathrm{cm}^{2}$? (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point.) (b) Estimate the maximum height from which a 70-kg man could jump and not fracture his tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s, and assume that the stress on his two legs is distributed equally.

A $15.0 \mathrm{~kg}$ mass fastened to the end of a steel wire with an unstretched length of $0.50 \mathrm{~m}$ is whirled in a vertical circle with angular velocity $2.00 \mathrm{rev} / \mathrm{s}$ at the bottom of the circle. The cross-sectional area of the wire is $0.010 \mathrm{~cm}^2$. Calculate the elongation of the wire when the mass is at the lowest point of the path.

In Fig. the upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both $50.0 \mathrm{~cm}$ long. The upper ball has mass $2.00 \mathrm{~kg}$, and it is initially $10.0 \mathrm{~cm}$ higher than the lower ball, which has mass $3.00 \mathrm{~kg}$. Find the frequency and maximum angular displacement of the motion after the collision.

A block with mass $M$ rests on a frictionless surface and is connected to a horizontal spring of force constant $k$. The other end of the spring is attached to a wall. A second block with mass $m$ rests on top of the first block. The coefficient of static friction between the blocks is $\mu_{\mathrm{s}}$. Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

Stress on a mountaineer's rope. A nylon rope used by mountaineers elongates $1.10 \mathrm{~m}$ under the weight of a $65.0 \mathrm{~kg}$ climber. If the rope is $45.0 \mathrm{~m}$ in length and $7.0 \mathrm{~mm}$ in diameter, what is Young's modulus for this nylon?

As the bob on a pendulum swings down toward its lowest point, its angular frequency $\omega$
A. increases.
B. decreases.
C. does not change.

An apple weighs 1.00 N. When you hang it from the end of a long spring of force constant 1.50 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back-and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

On the planet Newtonia, a simple pendulum having a bob with mass $1.25 \mathrm{~kg}$ and a length of $185.0 \mathrm{~cm}$ takes $1.42 \mathrm{~s}$, when released from rest, to swing through an angle of $12.5^{\circ}$, where it again has zero speed. The circumference of Newtonia is measured to be $51,400 \mathrm{~km}$. What is the mass of the planet Newtonia?

An object suspended from a spring vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic, and what fraction is potential?