Problem 46 - University Physics with Modern Physics

Question

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant \(k=450 \mathrm{N} / \mathrm{m}\) and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80 \(\mathrm{kg}\) and is to reach a maximum height of 3.6 \(\mathrm{m}\) above its initial position on the compressed spring, what distance must the bricklayer compress the spring initially? (The brick loses contact with the spring when the spring returns to its uncompressed length. Why?

Step-by-Step Solution

Answer

The spring must be compressed by approximately 0.5315 meters.

Step 1: Analyze the Problem

The problem involves a spring launching a brick upward. The goal is to find how much the spring must be compressed to propel the brick to a maximum height of 3.6 meters. We'll use the conservation of energy principle where the initial potential energy stored in the spring is converted into gravitational potential energy at the height.

Step 2: Write the Energy Conservation Equation

According to the conservation of energy, the potential energy stored in the spring when compressed is equal to the gravitational potential energy at the top of the brick's flight:\[ \frac{1}{2}kx^2 = mgh \]where \(k\) is the spring constant (450 N/m), \(x\) is the compression of the spring, \(m\) is the mass of the brick (1.80 kg), \(g\) is acceleration due to gravity (9.8 m/s²), and \(h\) is the height (3.6 m).

Step 3: Solve the Equation for x

Substitute the known values into the energy conservation equation:\[ \frac{1}{2} \cdot 450 \cdot x^2 = 1.80 \cdot 9.8 \cdot 3.6 \]First compute the right side:\(1.80 \cdot 9.8 \cdot 3.6 = 63.504\)Now the equation becomes:\[ 225x^2 = 63.504 \]

Step 4: Calculate Compression Distance x

Solve for \(x\) by dividing both sides by 225:\[ x^2 = \frac{63.504}{225} \]\[ x^2 = 0.28224 \]Take the square root of both sides:\[ x = \sqrt{0.28224} \approx 0.5315 \text{ meters} \]

Step 5: Conclusion

The bricklayer must compress the spring by approximately 0.5315 meters to propel the brick to a height of 3.6 meters. The brick loses contact with the spring once it returns to its original uncompressed length because there is no further force propelling the brick upward.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position or configuration. It is essentially the energy that has the potential to be converted into other forms of energy, like kinetic energy. The concept of potential energy is crucial in understanding energy transformation, especially in systems like springs and gravitational fields.

When a spring is compressed or stretched from its natural length, it stores energy. This stored energy is termed potential energy and can be calculated using the formula:

• \[ PE = \frac{1}{2} k x^2 \]

- In this equation, \(k\) represents the spring constant, and \(x\) indicates the distance the spring is compressed or stretched.

Understanding potential energy helps in solving problems related to energy conservation because it indicates how much energy can be transformed when the spring returns to its normal state.

Spring Constant

The spring constant, denoted by \(k\), is a measure of a spring's stiffness. It tells us how much force is needed to stretch or compress a spring by a certain amount. The spring constant is usually expressed in Newtons per meter (N/m).

A higher spring constant means the spring is stiffer, requiring more force to change its length, while a lower spring constant indicates a more flexible spring. In formula terms, the spring force can be calculated as:

• \[ F = kx \]

- Here, \(F\) represents the force applied or exerted by the spring, and \(x\) is the amount of compression or extension.

Knowing the spring constant is essential when dealing with systems involving mechanical energy, as it directly affects how much potential energy the spring can store.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. Specifically, it's the energy an object possesses because of its height above the ground or another reference point. The higher an object is positioned in the gravitational field, the more potential energy it has.

The formula for gravitational potential energy is:

• \[ PE_g = mgh \]

- In this equation: - \(PE_g\) is the gravitational potential energy - \(m\) is the mass of the object - \(g\) is the acceleration due to gravity (approximately \(9.8 \ m/s^2\) on Earth) - \(h\) is the height above the reference level.

Understanding gravitational potential energy is key to solving energy conservation problems involving objects moving vertically, like the brick in our exercise. It represents the energy the object will convert into other forms when it falls or moves upwards.

Mechanical Energy

Mechanical energy is the sum of potential energy and kinetic energy in a system. It is the total energy associated with the motion and position of an object.

In simple terms, mechanical energy allows an object to do work. For instance, when a spring is compressed, it stores potential energy, and upon release, the energy transforms into kinetic energy, propelling objects upward or outward.

In our exercise, the mechanical energy is initially stored as potential energy in the compressed spring, which is then converted to kinetic energy as the brick is propelled upward, reaching a maximum height where kinetic energy is zero and gravitational potential energy reaches its maximum.

• The principle of conservation states that the total mechanical energy remains constant in a closed system, excluding external forces like friction. Thus, \[ ext{Initial Mechanical Energy} = ext{Final Mechanical Energy} \]
• Understanding this concept is vital, as it helps predict the motion and energy distribution in systems involving forces like springs and gravity.