Table of contents

0. Math Review31m
- Math Review
  31m
1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

6. Intro to Forces (Dynamics)

Newton's First & Second Laws

Physics

6. Intro to Forces (Dynamics)

Newton's First & Second Laws

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6:13 minutes

Problem 62

Textbook Question

A 500 g ball moves horizontally with velocity v_𝓍 = ( 15 m) / (t + 1 s) for t > 0 s. What is the net force on the ball at t = 1 s?

Verified step by step guidance

Step 1: Identify the given quantities. The mass of the ball is m = 500 g = 0.5 kg (convert grams to kilograms). The velocity of the ball is given as vₓ = (15 m) / (t + 1 s). We are tasked with finding the net force on the ball at t = 1 s.

Step 2: Recall Newton's Second Law of Motion, which states that the net force on an object is given by F = m * a, where F is the net force, m is the mass, and a is the acceleration. To find the force, we first need to calculate the acceleration of the ball at t = 1 s.

Step 3: Acceleration is the time derivative of velocity. Differentiate the velocity function vₓ = (15 m) / (t + 1 s) with respect to time t. Use the quotient rule for differentiation: d(vₓ)/dt = d/dt[(15)/(t + 1)] = -15 / (t + 1)².

Step 4: Substitute t = 1 s into the expression for acceleration to find the acceleration at that specific time. The acceleration at t = 1 s is a = -15 / (1 + 1)² = -15 / 4 m/s².

Step 5: Use the formula F = m * a to calculate the net force. Substitute m = 0.5 kg and a = -15 / 4 m/s² into the equation. The net force is F = 0.5 * (-15 / 4) N. Simplify this expression to find the magnitude and direction of the net force.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law of Motion

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is crucial for determining the net force on the ball, as it relates the force to the object's mass and how its velocity changes over time.

Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. In this scenario, the velocity of the ball is given as a function of time, which means we need to differentiate this function to find the acceleration at a specific time, such as t = 1 s.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. Understanding kinematics is essential here, as it allows us to analyze the ball's velocity function and derive the necessary information about its motion to calculate the net force.

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Related Practice

Textbook Question

(II) A 3.0-kg object has the following two forces acting on it:

${\vec{F}}_{1} = (16 \hat{i} + 12 \hat{j}) N$

${\vec{F}}_{2} = (- 10 \hat{i} + 22 \hat{j}) N$

If the object is initially at rest, determine its velocity $\vec{v}$ at t = 4.0s .

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Multiple Choice

Which of the following statements about dynamics is correct?

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Multiple Choice

A box of mass ma is placed on a frictionless table. It is connected by a string to a hanging mass mb, as shown in the figure. The mass of the string can be neglected. The pulley is frictionless and has a negligible mass. What is the relative value of the acceleration of the masses?

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Multiple Choice

An object that weighs 75 N is pulled on a horizontal surface by a horizontal pull of 50 N to the right. The friction force on this object is 30 N to the left. What is the acceleration of the object?

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Textbook Question

A rocket traveling 1950 m/s away from the Earth at an altitude of 6400 km fires its rockets, which eject gas at a speed of 1200 m/s (relative to the rocket). If the mass of the rocket at this moment is 25,000 kg and an acceleration of 1.5 m/s² is desired, at what rate must the gases be ejected?

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Multiple Choice