Problem 53 - College Physics

Question

You've attached a bungee cord to a wagon and are using it to pull your little sister while you take her for a jaunt. The bungee's unstretched length is \(1.3 \mathrm{~m}\), and you happen to know that your little sister weighs \(220 \mathrm{~N}\) and the wagon weighs \(75 \mathrm{~N}\). Crossing a street, you accelerate from rest to your normal walking speed of \(1.5 \mathrm{~m} / \mathrm{s}\) in \(2.0 \mathrm{~s},\) and you notice that while you're accelerating, the bungee's length increases to about \(2.0 \mathrm{~m}\). What's the force constant of the bungee cord, assuming it obeys Hooke's law?

Step-by-Step Solution

Answer

The force constant of the bungee cord is 32.25 N/m.

Step 1: Identify the Problem and Known Values

We need to find the force constant (k) of the bungee cord, using Hooke's law. Given values include the unstretched length of the bungee (\(1.3 \mathrm{~m}\)), stretched length (\(2.0 \mathrm{~m}\)), little sister's weight (\(220 \mathrm{~N}\)), wagon's weight (\(75 \mathrm{~N}\)), final speed \(v_f = 1.5 \mathrm{~m/s}\), and time \(t = 2.0 \mathrm{~s}\).

Step 2: Determine the Total Force Needed for Acceleration

To calculate the force needed to accelerate the system, first find the total weight, which is the sum of your sister's weight and the wagon's weight: \(295 \mathrm{~N}\). Now, find the total mass \(m\) by dividing the total weight by gravity \(g = 9.8 \mathrm{~m/s^2}\): \(m = \frac{295 \mathrm{~N}}{9.8 \mathrm{~m/s^2}} = 30.1 \mathrm{~kg}\).

Step 3: Calculate the Acceleration

Using the formula \(a = \frac{v_f - v_i}{t}\), where \(v_i = 0\) (initial speed), calculate acceleration: \(a = \frac{1.5 \mathrm{~m/s} - 0}{2.0 \mathrm{~s}} = 0.75 \mathrm{~m/s^2}\).

Step 4: Calculate the Total Force Required for the Acceleration

Using Newton's second law \(F = ma\), calculate the force needed: \(F = 30.1 \mathrm{~kg} \times 0.75 \mathrm{~m/s^2} = 22.575 \mathrm{~N}\).

Step 5: Apply Hooke's Law to Find the Force Constant

Hooke's law is \( F = kx \), where \( F \) is the force applied, \( k \) is the force constant, and \( x \) is the extension in length. The extension \( x \) is \( 2.0 \mathrm{~m} - 1.3 \mathrm{~m} = 0.7 \mathrm{~m} \). Thus, \( k = \frac{22.575 \mathrm{~N}}{0.7 \mathrm{~m}} = 32.25 \mathrm{~N/m} \).

Key Concepts

Bungee cord physics

Bungee cords are fascinating tools often used to illustrate principles of physics. These cords, like elastic bands, stretch when a force is applied and return to their original shape when the force is removed. In our scenario, attaching a bungee cord to a wagon allows for an interesting application of physics principles.

• The bungee cord acts as a spring when stretched, storing potential energy that can later be converted to kinetic energy.
• The initial unstretched length of the cord is crucial since any extension will generate a restoring force.
• This restoring force is directly proportional to the amount that the bungee cord is stretched, revealing the core principle of Hooke's Law.

Understanding these aspects of bungee cord physics is key to solving problems where elasticity and stretching occur, like in our exercise.

Force constant calculation

The force constant, denoted as \( k \), is a measure of a spring’s stiffness, also applicable to bungee cords. In physics, determining \( k \) means assessing how resistive the cord is to being stretched.
Using Hooke's Law, the relationship between force and extension in springs can be given as:
\[ F = kx \]

• Where \( F \) is the force applied, \( k \) is the force constant of the bungee cord, and \( x \) is the extension of the cord beyond its original length.
• In our case, an extension from the resting length results from the acceleration applied to pull the wagon.
• The calculated force constant of 32.25 N/m indicates how much force per unit mass is required to extend the bungee by a meter.

Identifying the force constant is crucial in predicting how the cord will behave under various forces.

Newton's second law

Newton's second law of motion is a cornerstone concept in physics, stating:
\[ F = ma \]

• Here, \( F \) is the force applied to an object, \( m \) is the object's mass, and \( a \) is the acceleration produced.
• This law enables us to link kinetics and dynamics, impacting how we compute the forces involved in motion.
• In the context of pulling the wagon with your sister, Newton's second law allows us to calculate the total force needed to bring their mass to an accelerated state within a given timeframe.

Thus, understanding this law aids in understanding how forces result from changing speeds and directions.

Kinematics in physics

Kinematics concerns the motion of objects without considering the forces that cause the motion. In this problem, it connects to how the wagon moves and accelerates as you walk.
Key kinematic calculations include:

• Initial and final speeds—starting from rest (0 m/s) to reaching a walking speed of 1.5 m/s.
• Time duration—over 2.0 seconds to achieve the desired speed.
• Using these, the acceleration is determined by \( a = \frac{v_f - v_i}{t} \), computing an acceleration of 0.75 m/s².

Understanding kinematics provides insight into how objects transition between states of motion, vital for applying in diverse scenarios, from simple wagons to complex systems in physics.