/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 A 1450-kg submarine rises straig... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 70

A 1450-kg submarine rises straight up toward the surface. Seawater exerts both an upward buoyant force of \(16140 \mathrm{N}\) on the submarine and a downward resistive force of \(1030 \mathrm{N}\). What is the submarine's acceleration?

Short Answer

Expert verified
The submarine's acceleration is approximately \(0.62 \, \text{m/s}^2\) upwards.

Step by step solution

01

Identify Forces

The submarine is subject to three forces: the upward buoyant force (\(F_b = 16140 \, \text{N}\)), the downward resistive force (\(F_r = 1030 \, \text{N}\)), and the weight of the submarine (\(W = mg\)).
02

Calculate Weight

The weight of the submarine can be calculated using the formula \(W = mg\), where \(m = 1450 \, \text{kg}\) and \(g = 9.8 \, \text{m/s}^2\). \[ W = 1450 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 14210 \, \text{N} \]
03

Determine Net Force

To find the net force acting on the submarine, sum the forces considering their directions: \[ F_{net} = F_b - F_r - W \] Substitute the known values: \[ F_{net} = 16140 \, \text{N} - 1030 \, \text{N} - 14210 \, \text{N} = 900 \, \text{N} \]
04

Calculate Acceleration

Use Newton's second law to solve for acceleration: \(F_{net} = ma\). \[ a = \frac{F_{net}}{m} = \frac{900 \, \text{N}}{1450 \, \text{kg}} \approx 0.62 \, \text{m/s}^2 \]
05

Direction of Acceleration

Since the net force calculated is positive, the acceleration is in the upward direction, meaning the submarine is accelerating upwards.

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Key Concepts

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

Net Force
The concept of net force is crucial when analyzing the motion of objects, like our submarine. Net force is the overall force acting on an object after all the individual forces are summed up. To determine it, you need to consider both the magnitude and direction of each force acting on the object.
In the case of the submarine:
  • The upward buoyant force pushing the submarine up is 16140 N.
  • The downward resistive force opposing the motion of the submarine is 1030 N.
  • The weight of the submarine due to gravity is 14210 N.
By calculating the difference between these forces, we understand how they collectively influence the motion of the submarine.
The formula for net force (\( F_{net} \)) is: \[ F_{net} = F_b - F_r - W \] Substituting the known values gives us the net force as 900 N. A positive net force value here indicates a dominance of the forces moving the submarine upward.
Acceleration
Acceleration is the rate of change of velocity of an object over time. For the submarine, it tells us how quickly it's picking up speed as it moves upwards. We've determined the net force to be 900 N. To calculate acceleration, use Newton's second law:
\[ a = \frac{F_{net}}{m} \] Where \( F_{net} \) is the net force and \( m \) is the mass of the submarine, which is 1450 kg.
  • The resulting acceleration is approximately 0.62 m/s虏, signifying that the submarine gains speed at this rate every second as it rises towards the surface.
This signifies how much faster the submarine will be moving for each second it continues in motion. Understanding this helps in predicting how quickly the submarine will surface.
Newton's Second Law
Newton's second law is a fundamental principle that relates the net force acting on an object to its acceleration and mass. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The equation is given by: \[ F_{net} = ma \] This formula tells us that if any two quantities are known鈥攕uch as mass and net force鈥攚e can find the third, which in the submarine's case is acceleration.
  • In practical terms, when a net force is applied to an object, such as our rising submarine, it accelerates in the direction of the net force.
  • This relationship helps predict motion. If the net force was increased while mass stayed constant, the acceleration would also increase, causing the submarine to rise faster.
This law is fundamental for designing and predicting the behavior of moving objects in physics.

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Most popular questions from this chapter

The drawing shows Robin Hood (mass \(=77.0 \mathrm{kg}\) ) about to escape from a dangerous situation. With one hand, he is gripping the rope that holds up a chandelier (mass \(=195 \mathrm{kg}\) ). When he cuts the rope where it is tied to the floor, the chandelier will fall, and he will be pulled up toward a balcony above. Ignore the friction between the rope and the beams over which it slides, and find (a) the acceleration with which Robin is pulled upward and (b) the tension in the rope while Robin escapes.

An electron is a subatomic particle \(\left(m=9.11 \times 10^{-31} \mathrm{kg}\right)\) that is subject to electric forces. An electron moving in the \(+x\) direction accelerates from an initial velocity of \(+5.40 \times 10^{5} \mathrm{m} / \mathrm{s}\) to a final velocity of \(+2.10 \times 10^{6} \mathrm{m} / \mathrm{s}\) while traveling a distance of \(0.038 \mathrm{m} .\) The electron's acceleration is due to two electric forces parallel to the \(x\) axis: \(\overrightarrow{\mathbf{F}}_{1}=+7.50 \times 10^{-17} \mathrm{N},\) and \(\overrightarrow{\mathbf{F}}_{2},\) which points in the \(-x\) direction. Find the magnitudes of (a) the net force acting on the electron and (b) the electric force \(\overrightarrow{\mathbf{F}}_{2}\).

Scientists are experimenting with a kind of gun that may eventually be used to fire payloads directly into orbit. In one test, this gun accelerates a \(5.0-\mathrm{kg}\) projectile from rest to a speed of \(4.0 \times 10^{3} \mathrm{m} / \mathrm{s} .\) The net force accelerating the projectile is \(4.9 \times 10^{5} \mathrm{N}\). How much time is required for the projectile to come up to speed?

A bicyclist is coasting straight down a hill at a constant speed. The combined mass of the rider and bicycle is \(80.0 \mathrm{kg}\), and the hill is inclined at \(15.0^{\circ}\) with respect to the horizontal. Air resistance opposes the motion of the cyclist. Later, the bicyclist climbs the same hill at the same constant speed. How much force (directed parallel to the hill) must be applied to the bicycle in order for the bicyclist to climb the hill?

When a 58-g tennis ball is served, it accelerates from rest to a speed of \(45 \mathrm{m} / \mathrm{s}\). The impact with the racket gives the ball a constant acceleration over a distance of \(44 \mathrm{cm} .\) What is the magnitude of the net force acting on the ball?
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