Problem 57 - College Physics

Question

An "extreme" pogo stick utilizes a spring whose uncom- pressed length is 46 \(\mathrm{cm}\) and whose force constant is \(1.4 \times 10^{4} \mathrm{N} / \mathrm{m} . \mathrm{A} 60-\mathrm{kg}\) enthusiast is jumping on the pogo stick, compressing the spring to a length of only 5.0 \(\mathrm{cm}\) at the bottom of her jump. Calculate (a) the net upward force on her at the moment the spring reaches its greatest com- pression and (b) her upward acceleration, in \(\mathrm{m} / \mathrm{s}^{2}\) and \(g^{\prime}\) s at that moment.

Step-by-Step Solution

Answer

The net upward force is 5152 N, and the upward acceleration is approximately 85.87 m/s² or 8.76 g's.

Step 1: Understanding the Problem

We need to calculate two things: (a) the net upward force when the spring is compressed the most, and (b) the upward acceleration. Given: the relaxed spring length is 46 cm, the compressed spring length is 5 cm, the force constant (k) is \(1.4 \times 10^4 \, \mathrm{N/m}\), and the mass (m) is 60 kg.

Step 2: Calculate Displacement of the Spring

The displacement from the equilibrium position is the difference between the uncompressed and compressed lengths of the spring. It is calculated as follows: \( \Delta x = 46 \, \mathrm{cm} - 5 \, \mathrm{cm} = 41 \, \mathrm{cm} = 0.41 \, \mathrm{m} \).

Step 3: Calculate the Spring Force

Using Hooke's Law, the force exerted by the spring is \( F_s = k \Delta x \). Substituting the values, \( F_s = 1.4 \times 10^4 \, \mathrm{N/m} \times 0.41 \, \mathrm{m} = 5740 \, \mathrm{N} \).

Step 4: Calculate the Weight of the Enthusiast

The weight of the person can be calculated using the formula for gravitational force, \( F_g = mg \), where \( g = 9.8 \, \mathrm{m/s}^2 \). So, \( F_g = 60 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s}^2 = 588 \, \mathrm{N} \).

Step 5: Calculate the Net Upward Force

The net force is the difference between the spring force (upward) and the gravitational force (downward): \( F_{\text{net}} = F_s - F_g = 5740 \, \mathrm{N} - 588 \, \mathrm{N} = 5152 \, \mathrm{N} \).

Step 6: Calculate Upward Acceleration

Use Newton's second law, \( F_{\text{net}} = ma \), to solve for acceleration \( a \). \( a = \frac{F_{\text{net}}}{m} = \frac{5152 \, \mathrm{N}}{60 \, \mathrm{kg}} = 85.87 \, \mathrm{m/s}^2 \).

Step 7: Calculate Acceleration in Terms of g

Divide the acceleration by \( g \) to express it in terms of \( g \). \( a = \frac{85.87 \, \mathrm{m/s}^2}{9.8 \, \mathrm{m/s}^2} \approx 8.76 \, g \).

Key Concepts

Hooke's Law

Hooke's Law is a fundamental concept in physics that describes the force exerted by a spring when it is compressed or stretched. Named after Robert Hooke, the law is captured by the formula
\( F = k \Delta x \), where

• \( F \) is the force exerted by the spring,
• \( k \) is the spring constant (a measure of the stiffness of the spring),
• \( \Delta x \) is the displacement of the spring from its relaxed length.

This relationship tells us that the force exerted by a spring is proportional to its extension or compression.
In the pogo stick problem, the spring constant \( k \) is given as \( 1.4 \times 10^4 \, \mathrm{N/m} \), and the displacement \( \Delta x \) is \( 0.41 \, \mathrm{m} \). By applying Hooke's Law, the force exerted by the pogo stick spring at maximum compression is calculated to be \( 5740 \, \mathrm{N} \). Understanding this law helps in predicting how different springs will behave under various forces and is essential for solving problems involving spring mechanics.

Newton's Second Law

Newton's Second Law is a key principle that relates an object's mass, the net force acting upon it, and its acceleration.
The law is mathematically expressed as
\( F = ma \), where

• \( F \) is the net force applied to the object,
• \( m \) is the mass of the object,
• \( a \) is the acceleration of the object.

This equation shows that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
In the context of the pogo stick exercise, Newton's Second Law is used to determine the enthusiast's upward acceleration. With a calculated net force of \( 5152 \, \mathrm{N} \) acting on a mass of \( 60 \, \mathrm{kg} \), the acceleration is found to be \( 85.87 \, \mathrm{m/s^2} \). This result illustrates how variations in force and mass affect acceleration.

Spring Force Calculation

The spring force is an essential concept, especially in problems involving elastic materials like springs. It is calculated using Hooke's Law and represents the force exerted by a spring when it is either compressed or extended.
For the pogo stick problem, determining the spring force involves:

• Calculating the spring's displacement from its original length, which is \( 0.41 \, \mathrm{m} \).
• Using the spring constant \( 1.4 \times 10^4 \, \mathrm{N/m} \) and multiplying it by the displacement, \( \Delta x \).

This results in a spring force of \( 5740 \, \mathrm{N} \), which acts upwards against the gravitational force.
The clarity in calculating the spring force is crucial, as it directly impacts the net force and acceleration in mechanical systems.

Gravitational Force Calculation

In physics, the gravitational force calculation is a straightforward application of Newton's law of universal gravitation. It quantifies the force with which the Earth pulls an object towards its center.
The gravitational force is given by \( F_g = mg \), where

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \mathrm{m/s^2} \) on the surface of the Earth).

In our pogo stick scenario, this force is \( 588 \, \mathrm{N} \) for a mass of \( 60 \, \mathrm{kg} \).
Accurate calculation of gravitational force is important as it influences the net force and ultimately determines the motion of objects. Understanding how gravitational force combines with other forces like spring force is key to solving physics problems involving motion.