Question

Find where to place the fulcrum in a lever of length \(10 \mathrm{~m}\) so that a weight of \(14 \mathrm{~kg}\) at one end will balance a weight of \(10 \mathrm{~kg}\) at the other.

Short answer

Question: Calculate the position of the fulcrum in a lever to balance two weights of 14 kg and 10 kg, given that the lever is 10 meters long. Answer: The fulcrum must be placed at a distance of approximately \(4.17 \mathrm{~m}\) from the 14 kg weight in order to balance the lever.

Step-by-step solution

Write the moment equation for the lever balance.

In order to balance the lever, the moments of the two weights around the fulcrum must be equal and opposite. Let the distance from the 14 kg weight to the fulcrum be \(x\), so the distance from the 10 kg weight to the fulcrum must be \((10-x)\). The equation for the moments of the two weights is: moment of 14 kg weight = moment of 10 kg weight. \(14 \times x = 10 \times (10 - x)\)

Solve the moment equation for the distance x.

Now, we need to solve the equation for the distance, \(x\). \(14 \times x = 10 \times (10 - x)\) \(14x = 100 - 10x\) \(14x + 10x = 100\) \(24x = 100\) \(x = \frac{100}{24} = \frac{50}{12} = \frac{25}{6}\)

Calculate the position of the fulcrum.

The distance \(x\) is the distance from the 14 kg weight to the fulcrum. \(x = \frac{25}{6} \mathrm{~m}\) So, the fulcrum must be placed at a distance of \(\frac{25}{6} \mathrm{~m}\) or approximately \(4.17 \mathrm{~m}\) from the 14 kg weight in order to balance the lever.

Key concepts

Moment Equation

Imagine teeter-tottering on a playground seesaw: to balance, both sides must apply equal force over their respective distances from the central pivot point, or fulcrum. Similarly, in physics, the moment equation is critical for levers to find equilibrium. The moment is the product of the force applied and the distance from the fulcrum: \( Force \times Distance \).

Using the moment equation allows us to calculate the precise point at which a lever will balance. In your textbook problem, you have a 10-meter lever with weights of 14 kg and 10 kg on either end. By setting up the moment equation (14 kg \times Distance = 10 kg \times (10m - Distance)), we match the turning effect on both sides of the fulcrum. Solving this equation gives us the exact location to place the fulcrum to achieve balance.

Fulcrum Placement

The fulcrum is the pivoting point on which a lever rests and rotates. Its placement determines whether the lever will balance under different loads. Fulcrum placement is like finding the perfect spot on a beam to stand so that you don't tip over.

In a balanced lever, the total anti-clockwise moments and clockwise moments around the fulcrum are equal. If the fulcrum is not positioned correctly, the lever will rotate until it hits a stop or a new balance point is reached. As we determined from our moment equation, placing the fulcrum at \(\frac{25}{6} meters\) from the heavier weight effectively evens out the forces, illustrating the precision needed for proper fulcrum placement.

Physics of Levers

Levers are fascinating tools that can lift heavy objects with minimal effort, and they come in all shapes and sizes – from crowbars to bottle openers. The secret of their success lies in the physics of levers. They operate on the principle of moments, which states that the turning effect of a force is determined by the magnitude of the force and its distance from the pivot point or fulcrum.

There are three classes of levers, each with different fulcrum, effort, and load arrangements, yet all utlizing the same underlying physics. By adjusting distances from the fulcrum, you change the amount of force necessary to lift a weight. Your textbook problem focuses on a first-class lever, where the fulcrum is between the effort (force applied) and the load. It not only illustrates a physics principle but also showcases the practical application of simple machines.

Mechanical Equilibrium

When objects are at rest or move without acceleration, they're in mechanical equilibrium. For your lever problem, this means when the lever is not rotating, the moments on either side of the fulcrum are equal. This state of balance can happen for any number of forces, as long as they sum up to zero — which means there's no resultant force causing movement.

Mechanical equilibrium is like a perfectly calm see-saw with no ups and downs. It's not only essential in playground equipment but also in constructing buildings, designing furniture, and even in the human body - our muscles and bones act as levers! Maintaining equilibrium in the face of diverse and changing forces is a common and essential aspect of many systems, both in engineering and nature.