Rectilinear Motion II

  • Topics Covered

Dependent Motion and d’Alembert’s Principle

  • Dependent Motion and Related Problems

When motion of one body depends on the motion of other body (or bodies), it is known as dependent motion. Generally, these bodies are connected by a continuous, inelastic string.

An inelastic string does not change in length under the load, hence the total length of string always remains constant.

The objective of solving problems on dependent motion is to find the resulting motion of dependent bodies, when one of them is moved.

  • How to approach such problems?

We assume that all motions have at the most constant acceleration, i.e. \frac {da}{dt}=0

1) The very first decision one has to take is about the datum or a reference point. If the motions are vertical, a horizontal datum is chosen and if the motions are horizontal, a vertical datum is chosen. Note that for a system with both vertical and horizontal motions, 2 datums will be there.

If the motion is at angle, it is to be resolved vectorially in horizontal and vertical directions (with reference to the datum).

A sign convention is selected for each datum.

2) Now we make use of the fact that string is inextensible. We note the lengths of string connecting each member of the system and form an equation of the kind:

\sum_{i=1}^{n} x_i = L

Here, L denotes the total length of the string and x_i denotes the length of ith section of the string.

3) On differentiating above equation, we get the velocities and on further differentiating, we get the accelerations of each member.

Thus,

\sum_{i=1}^{n} v_i = 0

and

\sum_{i=1}^{n} a_i = 0

4) If there k different strings, there will be k equations for lengths.

  • Special Case – 2 Particles Connected by Single String

Suppose there are 2 particles A and B connected by a string. The arrangement is such that there are n_A parts connected to A and n_B parts are connected to B. Then,

\frac {n_A}{n_B}= \frac {x_A}{x_B} = \frac {v_A}{v_B}= \frac {a_A}{a_B}

  • D’Alembert’s Principle

The principle is named after the French mathematician Jean le Rond d’Alembert (1717-1783), who discovered it. It is a modified statement of Newton’s 2nd law of motion.

Consider an accelerating rigid body. This is a dynamic system. D’Alembert showed that the system can be converted into an equivalent static system by adding inertial force and inertial torque.

  • How is it used?

Consider a body of mass m, being pulled by a steel wire having tension T in it. If the block accelerates upwards with an acceleration a, we can write

\frac {T-mg}{m} = a

This is Newton’s 2nd law.

On rewriting,

T-mg=ma, \ i.e.\ T-mg + (-ma)=0

If a force equal to ma is added in opposite direction, the body appears to be under static equilibrium. This force is a fictitious force, called inertial force.

  • Application

It is easier to determine unknown forces on bodies, which are under static equilibrium than bodies which are moving. The analysis of machines and mechanisms gets tremendously simplified, when they are analyzed as static bodies, using d’Alembert’s principle.

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