Balancing an orange on a spoon experiment

We'll assume that your forefinger serves as the fulcrum or pivot point in what follows.

For the first case: The orange sits at the end of a lever arm. its weight times the length of the lever arm produces a torque which wants to twist the spoon counterclockwise around your forefinger.

To prevent this from occurring, you press your thumb down in contact with the end of the spoon handle to produce a counter-torque: the press-down force times the lever arm (which is the distance between the forefinger and the thumb- which is quite short in the first case) needs to cancel the torque produced by the orange. A short lever arm means high force, and you feel that in your fingers.

Second case: Here, the fulcrum (your forefinger) is closer to the orange, which gives the orange a shorter lever arm. the weight of the orange at the end of the lever arm hence produces less torque, and therefore your fingers can cancel the torque more easily. In addition, your thumb and forefinger are further apart, which lengthens the lever arm for your thumb so it produces more torque for the same force. both of these effects reduce the forces required to cancel the torque from the orange, and you feel the reduction in effort required.

You can model this system as a teeter-totter, with the orange on one end and your thumb on the other, with your forefinger as the pivot. Sliding the pivot towards the orange and away from your thumb simultaneously shortens the orange's lever arm while lengthening your thumb's lever arm. This lets your thumb balance the orange with less downforce.

Three forces are acting in this scenario. Two are downwards: the gravitational force on the orange and the force applied by your thumb. The third is upwards: the force applied by your forefinger. Since the scenario is static, the sum of all three forces is zero (counting for example the upward direction as positive and downward as negative).

$$ F_{orange} + F_{forefinger} + F_{thumb} = 0 $$

or equivalently:

$$ F_{thumb} = - (F_{orange} + F_{forefinger}) $$ The torque also needs to be zero, otherwise the spoon will start to rotate even when the sum of the forces is zero. Torque is defined as the distance to some chosen reference point times the force perpendicular to the line that connects to that reference point. The reference point can be chosen freely, so let's pick the orange as the reference point. The torque relative to this point is then:

$$ D_{forefinger}F_{forefinger} + D_{thumb}F_{thumb} = 0 $$

where $D_{forefinger}$ is the forefinger's distance to the orange, and $D_{thumb}$ is the thumb's distance to the orange. Since the orange has zero distance to itself, the $F_{orange}$ term has been omitted.

Solve this equation for $F_{thumb}$:

$$ F_{thumb} = - \frac{D_{forefinger}}{D_{thumb}} F_{forefinger} $$

Now we can eliminate $F_{thumb}$ and solve for $F_{forefinger}$:

$$ F_{forefinger} = - \frac{D_{thumb}}{D_{thumb}-D_{forefinger}} F_{orange} $$

Here we can see that the magnitude of $F_{forefinger}$ is equal to $F_{orange}$ when $D_{forefinger}$ is exactly zero, i.e. when the forefinger is placed directly under the orange. It gets larger the closer the forefinger is moved towards the thumb, since that makes the factor $\dfrac{D_{thumb}}{D_{thumb}-D_{forefinger}}$ grow. The magnitude of $F_{thumb}$ also grows since the sum of all three forces must be zero, and $F_{orange}$ is not changing.