Tutorials In Introductory Physics Answers

Mastering Introductory Physics: A Comprehensive Guide with Tutorial Answers

Introductory physics can feel daunting, a vast ocean of concepts and equations. This comprehensive guide aims to navigate you through those waters, providing detailed explanations and solutions to common tutorial problems. We'll cover key concepts, offer step-by-step solutions, and delve into the underlying physics principles. Whether you're struggling with kinematics, grappling with forces, or wrestling with energy conservation, this guide will equip you with the tools to succeed. Understanding these fundamentals is key to mastering more advanced physics topics.

1. Kinematics: The Language of Motion

Kinematics describes motion without considering the forces causing it. We'll focus on understanding displacement, velocity, and acceleration, along with their vector nature.

1.1. Displacement, Velocity, and Acceleration:

• Displacement (Δx): The change in position, a vector quantity. It's the final position minus the initial position. Note the importance of direction!
• Velocity (v): The rate of change of displacement. It's a vector, meaning it has both magnitude (speed) and direction. Average velocity is Δx/Δt; instantaneous velocity is the derivative of displacement with respect to time (dx/dt).
• Acceleration (a): The rate of change of velocity. It’s also a vector. Average acceleration is Δv/Δt; instantaneous acceleration is dv/dt or d²x/dt².

1.2. Equations of Motion (for constant acceleration):

These equations relate displacement, velocity, acceleration, and time when acceleration is constant.

• v = u + at (final velocity = initial velocity + acceleration × time)
• s = ut + ½at² (displacement = initial velocity × time + ½ × acceleration × time²)
• v² = u² + 2as (final velocity² = initial velocity² + 2 × acceleration × displacement)
• s = ½(u + v)t (displacement = ½ × (initial velocity + final velocity) × time)

Tutorial Problem 1: A car accelerates uniformly from rest to 20 m/s in 5 seconds. Calculate: (a) its acceleration; (b) the distance it travels during this time.

Solution:

(a) Using v = u + at, where u = 0 m/s (rest), v = 20 m/s, t = 5 s:

20 m/s = 0 m/s + a × 5 s

a = (20 m/s) / (5 s) = 4 m/s²

(b) Using s = ut + ½at², where u = 0 m/s, a = 4 m/s², t = 5 s:

s = 0 × 5 s + ½ × 4 m/s² × (5 s)² = 50 m

2. Forces and Newton's Laws of Motion

Newton's Laws form the bedrock of classical mechanics. They describe the relationship between forces and motion.

2.1. Newton's First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force.

2.2. Newton's Second Law (F = ma): The net force acting on an object is equal to the mass of the object multiplied by its acceleration. This is a vector equation; forces add vectorially.

2.3. Newton's Third Law (Action-Reaction): For every action, there is an equal and opposite reaction. These forces act on different objects.

2.4. Types of Forces: Gravity, friction, tension, normal force, etc., are common forces encountered in introductory physics.

Tutorial Problem 2: A 10 kg block is pushed across a frictionless surface with a force of 25 N. Calculate its acceleration.

Solution:

Using Newton's Second Law (F = ma):

25 N = 10 kg × a

a = (25 N) / (10 kg) = 2.5 m/s²

3. Energy and Work

Energy is the capacity to do work. Work is done when a force causes a displacement.

3.1. Work (W): The work done by a constant force is given by W = Fd cosθ, where F is the force, d is the displacement, and θ is the angle between the force and displacement vectors.

3.2. Kinetic Energy (KE): The energy of motion, given by KE = ½mv².

3.3. Potential Energy (PE): Stored energy. Gravitational potential energy is PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

3.4. Conservation of Energy: In a closed system, the total mechanical energy (KE + PE) remains constant.

Tutorial Problem 3: A 2 kg ball is dropped from a height of 10 m. Calculate its velocity just before it hits the ground, neglecting air resistance.

Solution:

Using conservation of energy:

Initial PE = mgh = 2 kg × 9.8 m/s² × 10 m = 196 J

Just before hitting the ground, all PE is converted to KE:

KE = ½mv² = 196 J

½ × 2 kg × v² = 196 J

v² = 196 m²/s²

v = 14 m/s

4. Momentum and Impulse

Momentum and impulse are crucial concepts in understanding collisions and changes in motion.

4.1. Momentum (p): The product of an object's mass and velocity: p = mv. It's a vector quantity.

4.2. Impulse (J): The change in momentum, equal to the average force multiplied by the time interval over which the force acts: J = FΔt = Δp.

4.3. Conservation of Momentum: In a closed system, the total momentum remains constant before and after a collision.

Tutorial Problem 4: A 0.5 kg ball moving at 10 m/s collides with a stationary 1 kg ball. After the collision, the 0.5 kg ball moves at 2 m/s in the opposite direction. Calculate the velocity of the 1 kg ball after the collision.

Solution:

Using conservation of momentum:

Initial momentum = 0.5 kg × 10 m/s + 1 kg × 0 m/s = 5 kg m/s

Final momentum = 0.5 kg × (-2 m/s) + 1 kg × v (where v is the velocity of the 1 kg ball)

Since momentum is conserved:

5 kg m/s = -1 kg m/s + 1 kg × v

v = 6 m/s

5. Rotational Motion

Rotational motion introduces new concepts like angular velocity, angular acceleration, and torque.

5.1. Angular Displacement (θ): The angle through which an object rotates.

5.2. Angular Velocity (ω): The rate of change of angular displacement: ω = Δθ/Δt.

5.3. Angular Acceleration (α): The rate of change of angular velocity: α = Δω/Δt.

5.4. Torque (τ): The rotational analogue of force: τ = rFsinθ, where r is the distance from the axis of rotation to the point where the force is applied, F is the force, and θ is the angle between the force vector and the lever arm.

Tutorial Problem 5: A constant torque of 10 Nm is applied to a wheel with a moment of inertia of 2 kg m². Calculate its angular acceleration.

Solution:

The rotational analogue of Newton's second law is τ = Iα, where I is the moment of inertia.

10 Nm = 2 kg m² × α

α = 5 rad/s²

6. Simple Harmonic Motion (SHM)

SHM describes oscillatory motion where the restoring force is proportional to the displacement from equilibrium.

6.1. Characteristics of SHM: Period, frequency, amplitude, displacement, velocity, and acceleration are key characteristics.

6.2. Equations of Motion for SHM: These often involve sinusoidal functions (sine and cosine).

Tutorial Problem 6: A mass on a spring oscillates with a period of 2 seconds. Calculate its frequency.

Solution:

Frequency (f) and period (T) are related by f = 1/T.

f = 1/2 s = 0.5 Hz

7. Waves and Sound

Waves transfer energy without transferring matter. Sound is a longitudinal wave.

7.1. Wave Properties: Wavelength, frequency, amplitude, speed, and intensity are important wave characteristics.

7.2. Sound Waves: Sound waves are characterized by their frequency (which determines pitch) and intensity (which determines loudness).

Tutorial Problem 7: A sound wave has a frequency of 440 Hz and a speed of 343 m/s. Calculate its wavelength.

Solution:

The speed of a wave (v) is related to its frequency (f) and wavelength (λ) by v = fλ.

343 m/s = 440 Hz × λ

λ = 0.78 m

8. Optics

Optics deals with the behavior of light. We'll cover reflection, refraction, and lenses.

8.1. Reflection: Light bounces off a surface. The angle of incidence equals the angle of reflection.

8.2. Refraction: Light bends as it passes from one medium to another. Snell's Law describes this phenomenon.

8.3. Lenses: Lenses use refraction to focus light. Converging lenses converge light, while diverging lenses diverge light.

Tutorial Problem 8: Light travels from air (n=1) into water (n=1.33) at an angle of 30 degrees to the normal. Calculate the angle of refraction.

Solution:

Using Snell's Law: n₁sinθ₁ = n₂sinθ₂

1 × sin(30°) = 1.33 × sin(θ₂)

θ₂ = sin⁻¹(sin(30°)/1.33) ≈ 22°

Frequently Asked Questions (FAQ)

• Q: What are the best resources for studying introductory physics? A: Textbooks, online courses (like Khan Academy, Coursera, edX), and practice problems are excellent resources.

• Q: How can I improve my problem-solving skills in physics? A: Practice consistently, break down problems into smaller steps, draw diagrams, and understand the underlying concepts.

• Q: What if I'm struggling with a particular concept? A: Seek help from your instructor, classmates, or online resources. Don't be afraid to ask for clarification.

• Q: Is it necessary to memorize all the formulas? A: While understanding the derivation is crucial, familiarity with key equations will streamline problem-solving. Focus on understanding how to apply them rather than rote memorization.

Conclusion

Introductory physics lays the foundation for a deeper understanding of the physical world. By mastering the concepts presented here, you’ll build a strong base for further studies in physics and related fields. Remember that consistent practice, a clear understanding of fundamental principles, and a willingness to seek help are crucial to success. Don’t be discouraged by initial challenges; with dedication and perseverance, you can conquer introductory physics and unlock the fascinating world of science. The journey may be challenging, but the rewards are immense. Keep exploring, keep questioning, and keep learning!