Contents
Physics Topics can be challenging to grasp, but the rewards for understanding them are immense.
What Does the Potential Energy Depend on?
Equilibrium of a Body Under Gravity
Suppose a chair is at rest with its legs resting on the floor. This stationary state of the chair ¡s its equilibrium position. If the chair is tilted a little, it has to be held in that position to prevent it from falling. This shows that this position is not the equilibrium position of the chair. On tilting it further the chair falls to the ground on its side and comes to rest again. This is another equilibrium position of the chair. In general, under the influence of gravity, a body may have one or more equilibrium states.
Under the influence of gravity, a body may stay in three types of equilibrium states:
Stable equilibrium: If a substance is disturbed from its equilibrium state, and if on release it gets back to the same equilibrium state, then the body is initially in a stable equilibrium. If a book, placed on a table with its largest surface resting on the table, is lifted a little by any one of the edges and then released, it falls back and returns to its initial state of equilibrium.
Unstable equilibrium: If a body, when displaced a little from its equilibrium position, tends to move away further, then the body was in an unstable equilibrium. Examples:
1. a pencil made to balance on its pointed end
2. an egg made to stand on its narrow end (It is almost impossible to keep a pencil or an egg in this position by hand) [Fig.]. If a slight push is given these fall over.
Neutral equilibrium: Let a body be displaced a little from its state of equilibrium and then released. If it still remains in equilibrium in its new position, then it is said to be in a neutral equilibrium. Let a sphere be placed on a smooth horizontal plane. On being pushed, it rolls a bit and comes to rest in a new position. Hence the sphere is said to be in a neutral equilibrium state.
Different equilibrium states of a body can be ascertained by either of the following two methods.
Extension of the Base
A chair on a floor rests with all four legs in contact with the floor. The quadrilateral obtained by joining the contact points of its legs with the floor is called the base of the chair. A body that has a larger base area compared to the area of its upper regions is always in stable equilibrium if the vertical line through its centre of gravity passes through its base.
From everyday experience we know that, for an object of a given height, stability increases with the extent of the base area. For a stable object, the line of action of its weight always lies within its base.
Stable equilibrium: Consider two possible ways of placing a brick on a horizontal surface [Fig(i)(a) and Fig.(i)(d)]. In Fig.(i)(a), the brick is placed on its largest face AB. This is a position of stable equilibrium and
the lines of action of the weight W and the normal reaction R coincide. If we tilt the brick to stand it on the edge through point A, as shown in Fig. (i)(b), the lines of action of W and R do not lie on the same straight line. Consequently, when we release the brick, the couple created by W and R brings it back on the original base AB. Note that the line of action of the weight W still passes through the base AB when the brick is tilted [Fig.(i)(b)]. So the initial state of the brick is a state of stable equilibrium.
To topple the brick, it needs to be tilted further about the edge through A so that the line of action of W goes outside the base AB. In Fig.(i)(c), the brick is on the verge of toppling.
In Fig.(i)(d), the same brick is shown to be standing on one of its smaller faces like BC. In this case, it will be easier to topple the brick because the line of action of W goes out side the base BC on turning the brick through a smaller angle. But this initial state is also a stable equilibrium state. Thus, we can say that stability increases with the extent of the base area.
Unstable Equilibrium: In Fig.(ii), a cone is shown balanced on its tip. Since the lines of action of W and R coincide, we can say that the cone is in equilibrium. However, a minute displacement from this position is enough to make it lose its balance and topple over. Thus, the cone standing on its tip is an example of unstable equilibrium. In this state, if the object is disturbed and then released, it does not return to its initial position but seeks a position of stable equilibrium. Note that when the cone is displaced slightly, the lines of action of W and R do not lie on the same straight line and the line of action of W does not pass through the base A. In Fig.(i)(c) and Fig.(i)(f), the bricks are at states of unstable equilibrium.
Neutral equilibrium: Consider a ball resting on a horizontal surface as shown in Fig.(iii). In this situation, the lines of action of W and R coincide. We conclude that the ball is in equilibrium. If we roll the ball to a new position, we find that it is in a similar kind of equilibrium as before. As a matter of fact, the lines of action of W and R coincide in any orientation. Such an equilibrium is called a neutral equilibrium. When a body is displaced from a position of neutral equilibrium it does not tend to return to its initial position; rather it settles to a new position of neutral equilibrium.
Thus, if the line of action of its weight does not pass through the base of the body, it cannot remain in equilibrium. Once displaced from its equilibrium position, a body comes to a new equilibrium either by returning to the old base or on a new base.
A chair is not stable when it stands on two legs, as the line of action of its weight does not pass through the base line defined by the two legs. Quadrupeds are more stable than animals who walk on two legs. Hence, in the latter case the technique of standing on two legs has to be learnt. When a bucket full of water is carried in the right hand, the body is leaned towards the left in order to keep the line of action of the weight of the body and the bucket together within our feet.
Position of Centre of Gravity
Objects tend to move towards the centre of the earth due to gravitational pull of the earth. Since the point of action of this attractive force (i.e., weight) is the centre of gravity of the body, the natural tendency of the centre of gravity is to attain the least height. Therefore, lower the position of the centre of gravity of a body, more stable its equilibrium.
If on displacing a body, its centre of gravity-
- goes up then the body was at a stable equilibrium initially,
- goes down, then the body attains a more stable equilibrium than before,
- remains at the same height, then the body is said to be in neutral equilibrium.
Example:
i) A person standing with his feet apart is more stable than when his feet are together. By placing his feet apart, the person increases the base area and so he can stand more comfortably. When a person lies down, along with the increase in the base area, there is a considerable lowering of the centre of gravity. Hence, the lying-down position is the most stable equilibrium for a person.
ii) Boxes and books, when placed on the floor, are usually kept with their largest surfaces in contact with the ground. This lowers the centre of gravity and the equilibrium becomes most stable.
iii) Passengers are not allowed to stand on the upper deck of a double decker bus in order to keep the centre of gravity low.
iv) Hydrometer, lactometer, special type of toys, etc. [Fig.] are loaded with lead in their lower part (base), so that the centre of gravity of the system remains very low. Hence, the vertical position is the most stable equilibrium position in these cases. The toys can be tilted about their bases through large angles, but they will come back to their vertical positions as soon as they are released. Hence, each of these toys has only one state of equilibrium.
Relationship of centre of gravity and gravitational potential with different types of equilibrium: Gravitational potential energy is related to the ‘position of centre of gravity of a body. Potential energy depends on the height of the centre of gravity from the ground. Lower the position of centre of gravity lower the potential energy and higher the stability of the body. Thus, when centre of gravity of a body or a system attains the lowest possible height, and hence has the lowest possible gravitational potential energy, the body achieves stable equilibrium. When centre of gravity is at its maximum height, potential energy is also the maximum and therefore, the equilibrium becomes unstable. The body tends to deviate from that position to a state of lower potential energy.
For a body in neutral equilibrium, the height of the centre of gravity does not change when it is displaced and the potential energy remains the same. Hence, the body remains in equilibrium even in the displaced state.