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To elucidate the assertion further, let us consider parabolic motion of a ball as shown in the figure. The important aspect of the parabolic motion is that the acceleration associated with motion is simply ‘g’ as there is no other force present except the force of gravity. The resultant force and mass of the ball together determine acceleration of the ball.
Problem : If the tension in the string is T, when the string makes an angle θ with the vertical. Find the acceleration of a pendulum bob, having a mass “m”.
Solution : As pointed out in our discussion, we need not study or consider velocities of bob to get the answer. Instead we should strive to know the resultant force to find out acceleration, using Newton’s second law of motion.
The forces, acting on the bob, are (i) force of gravity, mg, acting in the downward direction and (ii) Tension, T, acting along the string. Hence, the acceleration of the bob is determined by the resultant force, arising from the two forces.
Using parallelogram theorem for vector addition, the resultant force is :
$$\begin{array}{ll}F=\sqrt{\left\{{m}^{2}{g}^{2}+{T}^{2}+2mgT\mathrm{cos}(180\xb0-\theta )\right\}}=& \sqrt{\left({m}^{2}{g}^{2}+{T}^{2}+2mgT\mathrm{cos}\theta \right)}\end{array}$$
The acceleration is in the direction of force as shown in the figure, whereas the magnitude of the acceleration, a, is given by :
$$\begin{array}{l}a=\frac{\sqrt{\left({m}^{2}{g}^{2}+{T}^{2}+2mgT\mathrm{cos}\theta \right)}}{m}\end{array}$$
The change in velocity, resulting from the application of external force, may occur in magnitude or direction or both.
1: When force is applied in a given direction and the object is stationary.
Under this situation, the magnitude of velocity increases with time, while the body follows a linear path in the direction of force (or acceleration).
2: When force is applied in the direction of the motion, then it increases the magnitude of the velocity without any change in the direction.
Let us consider a block sliding on a smooth incline surface as shown in the figure. The component of the force due to gravity applies in the direction of motion. Under this situation, the magnitude of velocity increases with time, while the body follows a linear path. There is no change in the direction of motion.
3: When force is applied in the opposite direction to the motion, then it decreases the magnitude of the velocity without any change in the direction.
Take the example of a ball thrown vertically in the upward direction with certain velocity. Here, force due to gravity is acting downwards. The ball linearly rises to the maximum height till the velocity of ball reduces to zero.
During upward motion, we see that the force acts in the opposite direction to that of the velocity. Under this situation, the magnitude of velocity decreases with time, while the body follows a linear path. There is no change in the direction of motion.
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