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## ** 2. Cow Tipping (2 pts)** In 2005 Dr. Margo Lillie and her student Tracy Boechler published a study claiming that cow-tipping (the act of knocking a standing cow over by pushing on its side) cannot be performed by one person alone. You'll perform the same analysis here. Assume that a typical cow (see figure) has a mass of 680 kg, with a center of mass located as shown. Also assume that you push the cow horizontally at the point indicated in the figure. a) Draw a free-body diagram of the cow when a person is pushing it. b) Calculate the magnitude of the minimum force the tipper would need to exert to tip the cow. Do you think it would be reasonable for a single person to exert this force?

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Step 1:

To draw a free-body diagram of the cow when a person is pushing it, we need to identify and represent all the forces acting on the cow. The free-body diagram would look like this: \begin{align*} &{\text{Free-body diagram of the cow:}} \ &\begin{matrix} \begin{tikzpicture} \end{tikzpicture} \end{matrix} \end{align*} where CoG represents the center of gravity of the cow.

Step 2:

To calculate the magnitude of the minimum force the tipper would need to exert to tip the cow, we need to consider the torque generated by the force. The torque equation becomes: \begin{align*} \tau &= r \times F \times sin(\theta) \ r \times F_{min} \times sin(90^{\circ}) &= I \times \alpha \ r \times F_{min} &= I \times \alpha \ \end{align*} In this case, we have: \begin{align*} &= 410 \text{ kg} \cdot \text{m}^2 \end{align*} \begin{align*} \end{align*} When the cow starts tipping, it will rotate about its foot, and the rotational motion will continue until the cow's center of gravity passes over the foot. At that point, the cow will start to fall. We can find the angular acceleration by considering the time it takes for the center of gravity to pass over the foot. The angular acceleration can be found using the following formula: \begin{align*} \end{align*} We can find the linear velocity by considering the distance the center of gravity moves during the tipping process. The time it takes for the center of gravity to pass over the foot is given by: \begin{align*} \end{align*} Substituting the expression for acceleration into the angular acceleration formula, we get: \begin{align*} \end{align*} The time it takes for the center of gravity to pass over the foot can be found by considering the angular velocity of the cow's leg. The angular velocity is given by: \begin{align*} v &= \omega \cdot r \end{align*} Substituting the expression for linear velocity into the time formula, we get: \begin{align*} \end{align*} Substituting the expression for time into the angular acceleration formula, we get: \begin{align*} \end{align*} Substituting the expression for angular acceleration into the force formula, we get: \begin{align*} \end{align*} The angular velocity can be found by considering the angular displacement of the cow's leg. The angular displacement is given by: \begin{align*} s &= \theta \cdot r \end{align*} Therefore, the angular displacement is: \begin{align*} &= 2 \text{ rad} \end{align*} The angular velocity can be found by considering the time it takes for the center of gravity to pass over the foot. The angular velocity is given by: \begin{align*} \end{align*} Substituting the expression for angular velocity into the force formula, we get: \begin{align*} \end{align*} Substituting the known values, we get: \begin{align*} &= 343 \text{ N} \end{align*} Therefore, the minimum force required to tip the cow is 343 N. It is unlikely for a single person to exert this force, as it is approximately equal to 77 lbs.

Final Answer

The minimum force required to tip the cow is 343 N (approximately 77 lbs). It is unlikely for a single person to exert this force.

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