1.0 Turning Effects of Force and Equilibrium
In Class 9, we studied that a force can change the state of rest or motion of a body (Linear Motion). In Class 10, we advance to Rotational Motion. When a force is applied to a body which is pivoted at a point, the force tends to rotate the body about that point. This turning effect is a fundamental concept in mechanics.
Moment of a Force (Torque)
The Moment of a Force is equal to the product of the magnitude of the force and the perpendicular distance of the line of action of the force from the axis of rotation.
- S.I. Unit: Newton-metre ($N\,m$).
- C.G.S. Unit: dyne-cm ($1\,N\,m = 10^7 \text{ dyne-cm}$).
- Type: Vector quantity.
Moment of Force Formula
$$\tau = F \times d$$
Where: $\tau$ = Moment of Force, $F$ = Force applied, $d$ = Perpendicular distance from the pivot (Moment Arm).
1.1 Direction and Sign Convention
The direction of rotation depends on the point of application of force and the direction of the force. For numerical problems, we use the following convention:
- Anticlockwise Moment: Taken as Positive ($+$).
- Clockwise Moment: Taken as Negative ($-$).
A common mistake is taking the total length of the object as $d$. In Physics, $d$ is strictly the perpendicular distance from the axis of rotation to the line of action of force. If the force is applied at an angle, you must use the perpendicular component!
A nut is opened by a wrench of length $20\,cm$. If the least force required is $5\,N$, find the moment of force needed to turn the nut.
Solution:
1. Force ($F$): $5\,N$.
2. Distance ($d$): $20\,cm = 0.2\,m$ (Always convert to S.I. units).
3. Calculation: $\text{Moment of Force} = F \times d = 5 \times 0.2 = \mathbf{1\,N\,m}$.
Final Answer: The moment of force required is $1\,N\,m$.
Why is the handle of a door always provided at the free edge, as far as possible from the hinges? It is to maximize the distance ($d$), so that even a very small force ($F$) produces a large enough Moment to rotate the door easily!
2.0 Couples and the Principle of Moments
A single force applied to a pivoted body produces rotation, but in many practical scenarios, we use a Couple to produce a turning effect. A single force cannot produce pure rotation without causing some motion at the pivot; however, a couple allows for pure rotational motion.
What is a Couple?
Two equal and opposite parallel forces, not acting along the same line, form a Couple. The turning effect of a couple is called the Moment of the Couple.
- Example: Turning a steering wheel, opening a water tap, or turning a screwdriver.
- Key Property: The resultant force of a couple is zero, but the resultant moment is not zero.
Moment of a Couple
$$\text{Moment of Couple} = F \times d$$
Where: $F$ = Magnitude of either force, $d$ = Perpendicular distance between the two forces (Couple Arm).
2.1 Principle of Moments
For a body in equilibrium, the sum of all forces is zero, and the sum of all moments about any point is also zero. This leads us to the Principle of Moments, which is the backbone of most ICSE numericals in this chapter.
The Principle
According to the Principle of Moments, in equilibrium:
Sum of Anticlockwise Moments = Sum of Clockwise Moments
In many problems involving a uniform meter rule, students forget the weight of the scale itself. A uniform rule's weight always acts vertically downwards through its mid-point ($50\,cm$ mark). If the scale is pivoted elsewhere, this weight creates its own moment!
A uniform meter rule is balanced horizontally on a knife-edge placed at the $40\,cm$ mark when a weight of $20\,gf$ is suspended from the $10\,cm$ mark. Calculate the weight of the rule.
Solution:
1. Let $W$ be the weight of the rule acting at $50\,cm$. Pivot is at $40\,cm$.
2. Anticlockwise Moment: $20\,gf \times (40 - 10) = 20 \times 30 = 600\,gf\text{-}cm$.
3. Clockwise Moment: $W \times (50 - 40) = W \times 10 = 10W\,gf\text{-}cm$.
4. By Principle of Moments: $10W = 600 \Rightarrow W = \mathbf{60\,gf}$.
Final Answer: The weight of the rule is $60\,gf$.
When you use a screwdriver, your hand applies a Couple. The force from your palm and the opposing force from your fingers create equal and opposite forces on the handle, resulting in the pure rotation needed to drive the screw into wood.
3.0 Center of Gravity (C.G.)
Every body is made up of a large number of particles. The Earth attracts each particle towards its center with a force equal to its weight. Though these forces are many, they act as a single resultant force acting through a unique point. This point is known as the Center of Gravity.
What is Center of Gravity?
The Center of Gravity of a body is the point about which the algebraic sum of moments of weights of all the particles constituting the body is zero. The entire weight of the body is considered to act at this point.
- For a regular-shaped body, the C.G. lies at its geometric center.
- The C.G. does not necessarily have to be within the material of the body (e.g., a hollow ring).
3.1 C.G. of Common Regular Objects
For the ICSE Board Exam, you must memorize the position of C.G. for the following standard objects:
| Object | Position of C.G. |
|---|---|
| Rod | Mid-point of the rod. |
| Circular Disc | Geometric Center. |
| Solid Cone | On the axis, at height $h/4$ from the base. |
| Triangular Lamina | Centroid (Point of intersection of medians). |
Stability Condition
For a body to remain stable:
Vertical line from C.G. must fall within the base.
The lower the C.G. and the broader the base, the more stable the object.
Remember that the position of C.G. changes if the body is deformed. For example, if a straight wire is bent into a circle, its C.G. shifts from the mid-point of the wire to the center of the circle (which is outside the wire material).
Where does the Center of Gravity of a hollow cone lie compared to a solid cone of height 'h'?
Solution:
1. For a Solid Cone, the C.G. lies on the axis at a distance of $h/4$ from the base.
2. For a Hollow Cone, the C.G. lies on the axis at a distance of $h/3$ from the base.
3. Therefore, the C.G. of a hollow cone is higher than that of a solid cone of the same height.
Final Answer: Hollow cone C.G. = $h/3$; Solid cone C.G. = $h/4$.
Racing cars (like Formula 1 cars) are designed to be extremely low to the ground. This keeps their Center of Gravity very low, which prevents the car from toppling over even when taking sharp turns at incredibly high speeds.
4.0 Uniform Circular Motion
When a particle moves in a circular path with a constant speed, its motion is known as Uniform Circular Motion. Even though the speed is constant, the direction of motion changes at every point. Since velocity is a vector quantity, a change in direction means the velocity is changing, making this an accelerated motion.
Linear Motion vs. Circular Motion
- Uniform Linear Motion: Velocity is constant; Acceleration is zero.
- Uniform Circular Motion: Speed is constant; Velocity is variable; Acceleration is non-zero.
4.1 Centripetal and Centrifugal Force
To keep a body moving in a circle, a force must act on it. In ICSE Physics, distinguishing between the real force and the apparent force is crucial for conceptual clarity.
| Feature | Centripetal Force | Centrifugal Force |
|---|---|---|
| Direction | Towards the center. | Away from the center. |
| Nature | Real Force. | Pseudo (Fictitious) Force. |
Centripetal Force Formula
$$F = \frac{mv^2}{r}$$
Where: $m$ = mass of the body, $v$ = linear velocity, $r$ = radius of the circular path.
Centripetal and Centrifugal forces are NOT action-reaction pairs. An action-reaction pair acts on two different bodies. However, centrifugal force is a pseudo-force observed only in a rotating (non-inertial) frame of reference.
A stone of mass $100\,g$ is whirled in a horizontal circle of radius $2\,m$ at a constant speed of $10\,m/s$. Calculate the centripetal force acting on the stone.
Solution:
1. Mass ($m$): $100\,g = 0.1\,kg$.
2. Velocity ($v$): $10\,m/s$.
3. Radius ($r$): $2\,m$.
4. Calculation: $F = \frac{mv^2}{r} = \frac{0.1 \times (10)^2}{2} = \frac{0.1 \times 100}{2} = \frac{10}{2} = \mathbf{5\,N}$.
Final Answer: The centripetal force is $5\,N$.
The Earth is kept in its orbit around the Sun by the Gravitational Force, which provides the necessary Centripetal Force. Without this force, the Earth would fly off in a straight line, tangent to its orbit, into deep space!