8.5 Other force types  (Page 3/3)

However, we must note that force is transmitted undiminished under three important conditions : the string is (i) taut (ii) inextensible and (ii) mass-less. Unless otherwise stated, these conditions are implied when we refer string in the study of dynamics.

We must also appreciate that string is used with various combination of pulleys to effectively change the direction of force without changing the magnitude. There, usually, is a doubt in mind about the direction of tension in the string. We see that directions of tension in the same piece of string are shown in opposite directions.

However, this is not a concern that should be overemphasized. After all, this is the purpose of using string that force is communicated undiminished (T) but with change in direction. We choose the appropriate direction of tension in relation to the body, which is under focus for the study of motion. For example, the tension (T) is acting upward on the block, when we consider forces on the block. On the other hand, the tension is acting downward on the pulley, when we consider forces on the pulley. This inversion of direction of tension in a string is perfectly fine as tension work in opposite directions at any given intersection.

While considering string as element in the dynamic analysis, we should keep following aspects in mind :

1: If the string is taught and inextensible, then the velocity and acceleration of each point of the string (also of the objects attached to it) are same.

2: If the string is "mass-less", then the tension in the string is same at all points on the string.

3: If the string has certain mass, then the tension in the string is different at different points. If the distribution of mass is uniform, we account mass of the string in terms of "mass per unit length (λ)" - also called as linear mass density of the string.

4: If "mass-less" string passes over a "mass-less" pulley, then the tension in the string is same on two sides of the pulley.

5: If string having certain mass passes over a mass-less pulley, then the tension in the string is different on two sides of the pulley.

6: If "mass-less" string passes over a pulley with certain mass and there is no slipping between string and pulley, then the tension in the string is different on two sides of the pulley. Tensions of different magnitudes form the requisite torque required to rotate pulley of certain mass.

We have enlisted above scenarios involving string which are usually considered during dynamic analysis. Notably, we have not elaborated the reasons for each of the observations. We intend, however, to supplement these observations in appropriate context during the course.

Spring

Spring is a metallic coil, which can be stretched or compressed. Every spring has a “natural length” that can be measured, while the spring is lying on a horizontal surface (shown in the figure at the top).

If we keep one end fixed and apply a force at the other end to extend it as shown in the figure, then the spring stretches by a certain amount say Δx. In response to this, the spring applies an equal force in opposite direction to resist deformation (shown in the figure at the middle).

For a "mass-less" spring, it is experimentally found that :

where “k” is called spring constant, specific to a given spring. The negative sign is inserted to accommodate the fact that force exerted by spring is opposite to the direction of change in the length of spring. This relation is known as Hook's law and a spring, which follows Hook's law, is said to be perfectly elastic.

Similarly, when an external force compresses the spring, it opposes compression in the same manner (shown in the figure at the bottom).