Problem 17 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Two crates, of mass 65 \(\mathrm{kg}\) and 125 \(\mathrm{kg}\) , are in contact and at rest on a horizontal surface (Fig, \(32 ) . \mathrm{A} 650\) -N force is exerted on the 65 -kg crate. If the coefficient of kinetic friction is \(0.18,\) calculate \((a)\) the acceleration of the system, and \((b)\) the force that each crate exerts on the other. (c) Repeat with the crates reversed.

Step-by-Step Solution

Answer

(a) The acceleration is 1.66 m/s². (b) The force between the crates is 107.9 N (65 kg crate) or 207.5 N (125 kg crate). (c) The results are consistent when reversed.

Step 1: Calculate the Total Mass of the System

First, determine the total mass of the two crates. This is the sum of the individual masses: \( m_1 = 65 \, \mathrm{kg} \) and \( m_2 = 125 \, \mathrm{kg} \). Thus, the total mass \( m = m_1 + m_2 = 65 + 125 = 190 \, \mathrm{kg} \).

Step 2: Calculate the Normal Force and Friction Force

The normal force \( N \) for the entire system is equal to the gravitational force on both crates: \( N = m \cdot g = 190 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 1862 \, \mathrm{N} \). The friction force \( f_k \) is given by the equation \( f_k = \mu_k \cdot N \), where \( \mu_k = 0.18 \). So, \( f_k = 0.18 \times 1862 = 335.16 \, \mathrm{N} \).

Step 3: Determine Net Force on the System

The applied force on the system is 650 N. The net force \( F_{\text{net}} \) after considering friction is \( F_{\text{net}} = 650 \, \mathrm{N} - 335.16 \, \mathrm{N} = 314.84 \, \mathrm{N} \).

Step 4: Calculate the Acceleration of the System

Using Newton’s second law, \( F_{\text{net}} = m \cdot a \), solve for the acceleration \( a \): \( a = \frac{F_{\text{net}}}{m} = \frac{314.84}{190} \approx 1.66 \, \mathrm{m/s^2} \).

Step 5: Calculate the Force Between the Crates

To find the force that each crate exerts on the other, consider only one crate and isolate the forces acting on it. For the 65 kg crate, since it’s being pulled by 650 N, the net force \( F_{\text{net}1} \) for internal interaction can be found by subtracting the product of its mass and the previously calculated acceleration: \( F_{\text{int}} = 65 \, \mathrm{kg} \times 1.66 \, \mathrm{m/s^2} = 107.9 \, \mathrm{N} \).

Step 6: Repeat Calculation with Crates Reversed

Switch the position of the crates and follow similar steps for the crate starting with 125 kg with the applied force. Recap the calculations using the same acceleration due to the unchanged net force over total mass system value. Now, calculate internal force for the reversed position giving the force on the 125 kg crate which ensures consistency with both orientations due to same friction experienced: \( F_{\text{int}} = 125 \, \mathrm{kg} \times 1.66 \, \mathrm{m/s^2} = 207.5 \, \mathrm{N} \).

Key Concepts

Kinetic Friction

Kinetic friction plays a crucial role in this physics problem as it opposes the motion of the crates on a surface. When an object moves across a surface, kinetic friction acts in the opposite direction to slow down or prevent movement. In this exercise, the coefficient of kinetic friction is given as 0.18. This value helps in calculating the total frictional force acting against the movement of the crates.

The formula to determine the frictional force is:

• Friction Force ( \(f_k\)) = \(\mu_k \cdot N\), where \(\mu_k\) is the coefficient of kinetic friction and \(N\) is the normal force.

To find the normal force, we calculate the gravitational force, which is the weight of the combined masses of the crates pushing downwards due to gravity:

• Normal Force ( \(N\)) = \(m \cdot g\), where \(g = 9.8 \, \mathrm{m/s^2}\) (acceleration due to gravity).

For this problem, the gravitational force (and hence the normal force) was found to be 1862 N after adding the masses of the crates.

Finally, the frictional force needed to be subtracted from the applied force to find out the effective force moving the crates forward. By understanding kinetic friction, you can see how it affects the acceleration of objects in motion.

Acceleration Calculation

Acceleration is a measure of how quickly the velocity of an object changes with time. In this scenario, we want to determine how fast the two crates accelerate when a 650 N force is applied. Using Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration ( \(F = ma\)), we can find the acceleration of the system.

Initially, we calculated the total mass by adding the masses of both crates, which equals 190 kg. Then, we determined the net force applied to the system after accounting for the opposing force of kinetic friction (335.16 N). The net force is:

• Net Force ( \(F_{\text{net}}\)) = \(650 \, \mathrm{N} - 335.16 \, \mathrm{N} = 314.84 \, \mathrm{N}\).

We then applied the formula for acceleration ( \(a = \frac{F_{\text{net}}}{m}\)) to find the acceleration of the crates:

• Acceleration ( \(a\)) = \(\frac{314.84}{190} \approx 1.66 \, \mathrm{m/s^2}\).

By applying basic principles of physics, we see how the balance between applied forces and resistance affects the motion of objects. This makes it easy to predict how changes in forces would influence acceleration.

Interaction Forces

Interaction forces refer to forces that objects exert on each other when they are in contact. In this problem, the interaction forces come into play between the two crates. When the 65 kg crate is pushed, it exerts a force on the 125 kg crate, and vice versa.

To simplify, when the crates start to move as a unit, each experiences a reaction force from the other. You can calculate this force by considering the force resulting from the combined system acceleration affecting individual components.

For the 65 kg crate, the force it feels due to its interaction with the other crate is derived by calculating the force exerted based on the internal acceleration:

• Interaction Force = \(65 \, \mathrm{kg} \times 1.66 \, \mathrm{m/s^2} = 107.9 \, \mathrm{N}\).

When the crates are reversed, similar calculations can help determine the interaction force on the 125 kg crate:

• Interaction Force (reversed) = \(125 \, \mathrm{kg} \times 1.66 \, \mathrm{m/s^2} = 207.5 \, \mathrm{N}\).

These demonstrating calculations highlight the essence of interaction forces between objects in contact, emphasizing how internal system dynamics can alter external behavior.