Pho.rs: "Electro-static" analogies

Условие

Edu

Русский

Ohm's law describes a linear relationship between the current $I$ in a section of an electrical circuit and the voltage $U$ across that section: $I=\frac{1}{R} \cdot U$, where $R$ is the electrical resistance of that section.

Hooke's law is written similarly, according to which the elastic force $F$ arising during deformation of an elastic body (spring, rod, etc.) is proportional to the change in the length of the body $\Delta l$: $F = k \cdot \Delta l$, where $k$ is the coefficient of elasticity (stiffness) of the body. If $F > 0$ ($\Delta l > 0$), the spring is stretched. If $F < 0$ ($\Delta l < 0$), then the spring is compressed.

Consider a section of an electrical circuit consisting of three resistors with resistances $R_1$, $R_2$, $R_3$ (see Fig. 1). For node $B$, we write the relation for the currents: $I_1 = I_2 + I_3$. Also, the total voltage $U_0$ between points $A$ and $C$ is equal to the sum of the voltages: $U_0 = U_1 + U_2 = U_1 + U_3$.

Fig. 1

A similar form is exhibited by a mechanical system consisting of three light springs with stiffness coefficients $k_1$, $k_2$, and $k_3$, suspended from the ceiling (see Fig. 2). Assume that initially the springs in the system are undeformed. From the equilibrium condition for point $B$, one can write: $F_1 = F_2 + F_3$, where $F_1$, $F_2$, and $F_3$ are the elastic forces in the corresponding springs. The total deformation of the entire system $\Delta l_0$ can be represented as the sum of deformations: $\Delta l_0 = \Delta l_1 + \Delta l_2 = \Delta l_1 + \Delta l_3$, where $\Delta l_1$, $\Delta l_2$, and $\Delta l_3$ are the deformations of the corresponding springs.

Fig. 2

In what follows, when solving all parts of the problem, assume that the elements connecting the springs are light, non-deformable, and the system parameters are chosen so that these elements do not rotate under load (for example, they move along smooth vertical guides), the springs remain vertical under any deformation and are never compressed so much that the coils are tightly pressed against each other.

Part A. "Simple" system (2.0 points)

The equations describing the electrical circuit and the mechanical system are analogous. For example, the inverse stiffness of a spring $1/k$ in the mechanical system is a quantity analogous to electrical resistance $R$ in the electrical circuit.

A1 ^1.00 A force $F_1$ is applied to point $C$ (Fig. 2), directed vertically downward. What is the deformation $\Delta l_2$ of the spring with stiffness coefficient $k_2$ in this case? Express the answer in terms of $k_1$, $k_2$, $k_3$, and $F_1$.

A2 ^1.00 Determine the effective stiffness coefficient $k_0$ of the spring system $AC$ shown in Fig. 2. Express the answer in terms of $k_1$, $k_2$, and $k_3$.

Note. The effective stiffness coefficient of the system in this case is the ratio of the magnitude of the force applied to point $C$ to the magnitude of the displacement of the point of application of this force.

Part B. "Symmetric" system (3.5 points)

The system shown in Fig. 3 consists of identical light springs with stiffness coefficients $k$. The springs are initially undeformed.

Fig. 3

B1 ^1.00 Determine the effective stiffness $k_E$ of the system if a force directed vertically downward is applied to point $E$. The answer must be expressed in terms of $k$.

B2 ^2.50 Determine the effective stiffness $k_Y$ of the system if a force directed vertically downward is applied to point $Y$. The answer must be expressed in terms of $k$.

Part C. "Inifinte" system (4.5 points)

The system shown in Fig. 4 consists of an infinite number of light springs with identical stiffness coefficients $k$. The springs are initially undeformed.

Fig. 4

C1 ^1.50 Determine the effective stiffness $k_\infty$ of the system if a force directed vertically downward is applied to point $H$. The answer must be expressed in terms of $k$.

C2 ^1.50 Determine the deformation $\Delta l_x$ of spring "$x$" if two identical forces $F$, directed vertically downward, are applied to points $G$ and $H$. The answer must be expressed in terms of $F$ and $k$.