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Chapter 5: Problem 12

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

Short Answer

Expert verified
The initial acceleration of the rocket is \(3.96 \mathrm{~m/s^2}\) and the force exerted by the floor on the power supply is \(21.76 \mathrm{~N}\)

Step by step solution

01

Calculate the Initial Acceleration

Firstly, calculate the net force acting on the rocket. This is equal to the thrust force minus the gravitational force. The gravitational force can be calculated by multiplying the mass of the rocket with the gravity constant, which is approximately \(9.8 \mathrm{~m/s^2}\). So, the gravitational force is \(125 \mathrm{~kg} * 9.8 \mathrm{~m/s^2} = 1225 \mathrm{~N}\). After subtracting the gravitational force from the thrust, the net force will be \(1720 \mathrm{~N} - 1225 \mathrm{~N} = 495 \mathrm{~N}\). Using Newton's second law, the initial acceleration \(a\) is given by \(a = F/m = 495 \mathrm{~N} / 125 \mathrm{~kg} = 3.96 \mathrm{~m/s^2}\)
02

Calculate the Force Exerted by the Floor on the Power Supply

The force exerted by the floor on the power supply while the rocket is accelerating is the weight of the power supply plus the additional acceleration due to the rocket's acceleration. The weight of the power supply under normal conditions can be calculated by dividing the given weight (15.5 N) by the gravity constant (9.8 m/s^2), which gives the mass as approximately 1.58 kg. Therefore, the force exerted by the floor on the power supply is given by \(F = m * (g + a) = 1.58 \mathrm{~kg} * (9.8 \mathrm{~m/s^2} + 3.96 \mathrm{~m/s^2}) = 21.76 \mathrm{~N}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thrust Force
Thrust is a crucial concept in understanding how rockets propel themselves off the ground. It's a mechanical force, so despite it often being associated with rockets, it's not exclusive to them. Created by the rocket's engine, thrust moves the rocket in the opposite direction to the exhaust gases shooting out the back. The force is produced as a result of Newton鈥檚 third law, which tells us for every action, there's an equal and opposite reaction.

In the case of our textbook exercise, the rocket generates a constant thrust force of 1720 N. This force is what's primarily responsible for overcoming the gravitational pull of the earth and allowing the rocket to ascend.
Gravitational Force
Everything that has mass is pulled by gravity toward the center of the Earth. This pull is what we call the gravitational force. It's also known as weight. Now, the more mass an object has, the stronger the gravitational force acting on it. For any object on or near the surface of Earth, this force is calculated by multiplying the object's mass by Earth's acceleration due to gravity, which is approximately 9.8 m/s虏.

In our example, the rocket with all its contents weighs 125 kg, resulting in a gravitational force of 1225 N pulling it down. This is an essential value to consider when calculating the net force for the rocket's acceleration.
Newton's Second Law
One of the cornerstone concepts of physics, Newton's second law, tells us that the acceleration of an object depends directly on the net forces acting upon it and inversely on its mass. The law is succinctly captured by the equation: \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration. In our exercise, we apply this law to find the rocket's initial acceleration by dividing the net force (thrust minus gravitational force) by the mass of the rocket.
Free-Body Diagram
Free-body diagrams are instrumental in visualizing forces acting upon an object. They are simple representations that pinpoint all the forces applied to an object, which helps us understand how these forces interact. In our exercise, drawing a free-body diagram for the power supply inside the rocket clarifies how the gravitational force and the rocket鈥檚 acceleration influence it. By plotting out the forces, students can more easily calculate the net force -- and therefore, the acceleration -- acting on the power supply.
Rocket Mass
The mass of an object is a measure of how much matter it contains. Unlike weight, mass does not change with location, even when the object is on the moon or in deep space. In the context of our exercise, the mass of the rocket is pivotal in determining both the gravitational force acting on it and its acceleration. The initial mass of 125 kg is used to calculate the force of gravity pulling it towards Earth, and to find the net force for acceleration once the thrust is applied.
Acceleration Due to Gravity
Commonly symbolized as \( g \), the acceleration due to gravity at Earth's surface is approximately 9.8 m/s虏. It's what gives weight to objects and causes them to fall towards the ground when dropped. This constant is a fundamental ingredient in calculating weight (gravitational force) and is vital for understanding the conditions under which the rocket operates on Earth. In our scenario, the power supply's weight and the additional force from the rocket's acceleration are both influenced by this value.

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Most popular questions from this chapter

A \(3.00 \mathrm{~kg}\) box that is several hundred meters above the earth's surface is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope and results in a tension in the rope of \(T(t)=(36.0 \mathrm{~N} / \mathrm{s}) t\) The box is at rest at \(t=0 .\) The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) \(t=1.00 \mathrm{~s}\) and (ii) \(t=3.00 \mathrm{~s} ?\) (b) What is the maximum distance that the box descends below its initial position? (c) At what value of \(t\) does the box return to its initial position?

A steel ball with mass \(m\) is suspended from the ceiling at the bottom end of a light, 15.0 -m-long rope. The ball swings back and forth like a pendulum. When the ball is at its lowest point and the rope is vertical, the tension in the rope is three times the weight of the ball, so \(T=3 m g .\) (a) What is the speed of the ball as it swings through this point? (b) What is the speed of the ball if \(T=m g\) at this point, where the rope is vertical?

When a crate with mass \(25.0 \mathrm{~kg}\) is placed on a ramp that is inclined at an angle \(\alpha\) below the horizontal, it slides down the ramp with an acceleration of \(4.9 \mathrm{~m} / \mathrm{s}^{2} .\) The ramp is not friction less. To increase the acceleration of the crate, a downward vertical force \(\overrightarrow{\boldsymbol{F}}\) is applied to the top of the crate. What must \(F\) be in order to increase the acceleration of the crate so that it is \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) ? How does the value of \(F\) that you calculate compare to the weight of the crate?

Two blocks are suspended from opposite ends of a light rope that passes over a light, friction less pulley. One block has mass \(m_{1}\) and the other has mass \(m_{2},\) where \(m_{2}>m_{1}\). The two blocks are released from rest, and the block with mass \(m_{2}\) moves downward \(5.00 \mathrm{~m}\) in \(2.00 \mathrm{~s}\) after being released. While the blocks are moving, the tension in the rope is \(16.0 \mathrm{~N}\). Calculate \(m_{1}\) and \(m_{2}\).

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius \(2.5 \mathrm{~m} .\) The cylinder started to rotate, and when it reached a constant rotation rate of \(0.60 \mathrm{rev} / \mathrm{s},\) the floor dropped about \(0.5 \mathrm{~m}\). The people remained pinned against the wall without touching the floor. (a) Draw a force diagram for a person on this ride after the floor has dropped. (b) What minimum coefficient of static friction was required for the person not to slide downward to the new position of the floor? (c) Does your answer in part (b) depend on the person's mass? (Note: When such a ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the wall to the floor.
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