Theoretical Mechanics IPSP

Problem 5.11: Determining the volume, the mass, and the center of mass
Determine the mass $M$, the area or volume $V$, and the center of mass $\mathbf Q$ of bodies with the following mass density and shape.

1. A triangle in two dimensions with constant mass density $\rho = 1 \, \text{kg/m$^2$}$ and side length $6 \, \text{cm}$, $8 \, \text{cm}$, and $10 \, \text{cm}$. Hint: Determine first the angles at the corners of the triangle. Decide then about a convenient choice of the coordinate system (position of the origin and direction of the coordinate axes).
2. A circle in two dimensions with center at position $(a,b)$, radius $R = 5 \, \text{cm}$, and constant mass density $\rho = 1 \, \text{kg/m$^2$}$. Hint: How do $M$, $V$ and $\mathbf Q$ depend on the choice of the origin of the coordinate system?
3. A rectangle in two dimensions, parameterized by coordinates $0 \leq x \leq W$ and $0 \leq y \leq B$, and a mass density $\rho(x,y) = \alpha \, x$. What is the dimension of $\alpha$ in this case?
4. A three-dimensional wedge with constant mass density $\rho = 1 \, \text{kg/m$^3$}$ that is parameterized by $0 \leq x \leq W$, $0 \leq y \leq B$, and $0 \leq z \leq H - H x/W$. Discuss the relation to the result of part 2).
5. A cube with edge length $L$. When its axes are aligned parallel to the axes $\hat x, \hat y$, $\hat z$, it density takes the form $\rho(x,y,z) = \beta \, z$. What is the dimension of $\beta$ in this case?

Problem 5.12: Return time and position of the professor

1. How long will the professor take to arrive in down-under, and when will he reappear for the first time close to home?
2. How far will Earth have moved in that time? When this happens to him in Leipzig, where will he reappear, and when will he see land again for the next time?
3. Adopt an orthonormal coordinate system $(x,y,z)$ that is co-rotating with Earth, with origin in the Earth center, $z$-axis oriented towards the North pole, and $x$-direction towards the latitude of Leipzig. Sketch the trajectory of the professor in the $(x,y)$-plane when he was at rest initially.
4. Observe that the professor is initially standing on the surface of Earth. What does this imply for his initial velocity? How does the trajectory change?
5. Let him how start with zero velocity from the Moon surface. What does this imply for the force law? How does the trajectory change?