Problem 12 - University Physics with Modern Physics

Question

A 125 -kg (including all the contents) rocket has an engine that produces a constant vertical force (the thrust) of 1720 \(\mathrm{N}\) . Inside this rocket, a 15.5 -N electrical power supply rests on the floor. (a) Find the acceleration of the rocket. (b) When it has reached an altitude of \(120 \mathrm{m},\) how hard does the floor push on the power supply? (Hint: Start with a free-body diagram for the power supply.)

Step-by-Step Solution

Answer

(a) Rocket's acceleration: 3.96 m/s². (b) Floor's force on power supply: 21.75 N.

Step 1: Calculate Net Force

First, identify the forces acting on the rocket. The thrust force is 1720 N upward, and the gravitational force is acting downward. Calculate the gravitational force using the formula \( F_{gravity} = m imes g \), where \( m = 125 \) kg (mass of the rocket) and \( g = 9.8 \) m/s² (acceleration due to gravity). \[ F_{gravity} = 125 \times 9.8 = 1225 \text{ N} \] The net force is the thrust force minus the gravitational force: \( F_{net} = 1720 - 1225 = 495 \text{ N} \).

Step 2: Calculate Rocket's Acceleration

Use Newton’s second law to find the rocket's acceleration: \( F_{net} = m imes a \).Rearrange the formula to solve for acceleration \( a \): \( a = \frac{F_{net}}{m} \). Using \( F_{net} = 495 \text{ N} \) and \( m = 125 \text{ kg} \): \[ a = \frac{495}{125} = 3.96 \text{ m/s}^2 \].

Step 3: Analyze Free-Body Diagram for the Power Supply

Consider the forces acting on the 15.5 N electrical power supply. The forces include the normal force \( N \) (floor's push on the power supply) and gravitational force \( F_g = 15.5 \text{ N} \) (downward). The power supply is inside the rocket, moving with acceleration \( a = 3.96 \text{ m/s}^2 \).

Step 4: Apply Newton’s Second Law on Power Supply

For the power supply: \( N - F_g = m_{ps} \times a \), where \( m_{ps} \) is the mass of the power supply.First, calculate \( m_{ps} \) using \( F_g = m_{ps} imes g \): \( m_{ps} = \frac{15.5}{9.8} \approx 1.58 \text{ kg} \).Substitute \( m_{ps} \) and \( a \) into the equation: \[ N - 15.5 = 1.58 \times 3.96 \] \[ N = 6.253 + 15.5 \] \[ N = 21.753 \text{ N} \].

Step 5: Conclusion

The acceleration of the rocket is \( 3.96 \text{ m/s}^2 \). At an altitude of 120 m, the floor of the rocket exerts a force of about \( 21.75 \text{ N} \) on the power supply.

Key Concepts

Rocket Propulsion

Rocket propulsion is an intriguing concept that applies Newton's Second Law of Motion, which states that the acceleration of an object depends on the net force acting upon it and the object's mass. For a rocket, the engine produces a thrust force that propels it vertically upward.
As the rocket travels upward, this thrust has to overcome the gravitational force that pulls it downward to achieve lift-off. This is mathematically represented as:

• Thrust force (upward) and gravitational force (downward) oppose each other.
• The net force, which determines the rocket's movement, is the difference between these forces.

The thrust force accelerates the rocket, allowing it to ascend into the sky by producing a force greater than the gravitational pull. This net upward force results in the acceleration calculated from the formula, a central component of rocket science theory.
Understanding thrust and rocket propulsion involves realizing how engines are engineered to exert greater force to compensate for the rocket’s mass and the constant force of gravity.

Free-Body Diagram

A free-body diagram is a useful tool in physics to visualize the different forces acting on an object. In this case, the diagram helps us understand how the forces interact on the power supply inside the rocket.
To draw a free-body diagram for the power supply, pay attention to:

• The downward gravitational force, which in this scenario is 15.5 N.
• The normal force, which is the force exerted by the floor of the rocket on the power supply to support it in the upward acceleration.

The power supply isn't accelerating independently; instead, it moves with the same acceleration as the rocket, which we calculated to be 3.96 m/s².
This method allows us to determine how much force the floor of the rocket exerts on the power supply, termed the normal force, which is crucial for understanding interaction forces within accelerating systems.

Gravity Force Calculation

Calculating the force due to gravity is essential when analyzing any movement, especially in a vertical motion like that of a rocket. The gravitational force is how much the Earth pulls on an object and is measured using the equation:

• \( F_{gravity} = m \times g \)
• where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.

In our rocket exercise, we considered two gravitational force calculations:

• One for the rocket, weighing 125 kg, which calculated to be 1225 N.
• Another for the electrical power supply inside, weighing about 1.58 kg, resulting in a 15.5 N gravitational pull.

These calculations are significant as they allow us to distinguish the forces that need to be overcome for the rocket to climb upwards. Understanding gravity's role helps us accurately compute other forces and anticipate movement dynamics in physics and engineering scenarios.