Question

The maximum force you can exert on an object with one of your back teeth is about 750 N. Suppose that as you gradually bite on a clump of licorice, the licorice resists compression by one of your teeth by acting like a spring for which $\mathit{k}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathit{N}\mathbf{/}\mathit{m}$. Find (a) the distance the licorice is compressed by your tooth and (b) the work the tooth does on the licorice during the compression. (c) Plot the magnitude of your force versus the compression distance. (d) If there is a potential energy associated with this compression, plot it versus compression distance. In the 1990s the pelvis of a particular Triceratops dinosaur was found to have deep bite marks. The shape of the marks suggested that they were made by a Tyrannosaurus rex dinosaur. To test the idea, researchers made a replica of a T. rex tooth from bronze and aluminum and then used a hydraulic press to gradually drive the replica into cow bone to the depth seen in the Triceratops bone. A graph of the force required versus depth of penetration is given in Fig. 8-71 for one trial; the required force increased with depth because, as the nearly conical tooth penetrated the bone, more of the tooth came in contact with the bone. (e) How much work was done by the hydraulic press—and thus presumably by the T. rex—in such a penetration? (f) Is there a potential energy associated with this penetration? (The large biting force and energy expenditure attributed to the T. rex by this research suggest that the animal was a predator and not a scavenger.)

Short answer

1. The distance the licorice is compressed by your tooth is 0.003 m
2. Work done by tooth on licorice is 1.1J
3. The magnitude of force versus the compression distance is plotted.
4. The potential energy versus the compression distance is plotted.
5. Work done by hydraulic press is 48 J
6. There is no potential associated with penetration.

Step-by-step solution

The given data

1. Maximum force you can exert on an object is F=750N
2. Spring stiffness is$k=2.5\times {10}^{5}N/m$

Understanding the concept of spring force

We can find displacement using the formula for the spring force in terms of spring constant and displacement. Using the formula for the work done in terms of restoring force and displacement, we can find the work done by the spring as well.

Formulae:

The force applied on a spring, F=kx (i)

The work done by the force on the spring, $W=0.5k{x}^2$ (ii)

a) Calculation of the distance the licorice is compressed by tooth

Using equation (i), the compression length of the licorice is given as:

$$\begin{gathered} 750=2.5\times {10}^5\times x \\ x=0.003m \end{gathered}$$

Hence, the value of the distance is 0.003 m.

b) Calculation of the work done by the licorice on the tooth

Work done by the licorice on the tooth is calculated using equation (ii) as follows:

$$\begin{gathered} W=0.5\times 2.5\times {10}^5\times 0.{003}^3 \\ =1.1J \end{gathered}$$

Hence, the value of the work is 1.1J.

c) Plotting the graph between force and compressed distance

Plot of magnitude of force versus compression distance

Hence, the required graph is plotted.

d) Plotting the graph between potential energy and compressed distance

Plot of potential energy versus compression distance

Hence, the graph required for potential energy is plotted.

e) Calculation of the work done by the hydraulic press

Work done by hydraulic forceis the areaunder the given force-distance curve. That is given as follows:

$$\begin{gathered} W=0.5\times 8000\times 0.012 \\ =48J \end{gathered}$$

Hence, the value of the work done is 48 J.

f) Calculation of potential energy dependence with penetration

From the graph plotted by the potential energy in part (d), we can see that the energy depends on the compression distance.

There is no potential energy associated with penetration.