Mechanics
Sample Problem 1: Kinematics
A car accelerates uniformly from rest to a speed of 25 m/s over a distance of 100 meters. Calculate the time taken for the car to reach this speed.
Solution:
To find the time taken, we can use the kinematic equation:
\[
v^2 = u^2 + 2as
\]
Where:
- \(v = 25 \text{ m/s}\) (final velocity)
- \(u = 0 \text{ m/s}\) (initial velocity)
- \(s = 100 \text{ m}\) (distance)
- \(a\) is the acceleration.
First, rearranging the equation to solve for acceleration \(a\):
\[
25^2 = 0 + 2a(100)
\]
\[
625 = 200a
\]
\[
a = \frac{625}{200} = 3.125 \text{ m/s}^2
\]
Now, using the formula for time:
\[
v = u + at
\]
We can rearrange to solve for \(t\):
\[
t = \frac{v - u}{a} = \frac{25 - 0}{3.125} = 8 \text{ seconds}
\]
Final Answer: The time taken for the car to reach the speed of 25 m/s is 8 seconds.
Sample Problem 2: Newton's Laws
A 5 kg block is resting on a horizontal surface. The coefficient of friction between the block and the surface is 0.2. What is the maximum force of friction acting on the block?
Solution:
The force of friction can be calculated using the formula:
\[
f_f = \mu N
\]
Where:
- \(f_f\) is the force of friction,
- \(\mu = 0.2\) (coefficient of friction),
- \(N\) is the normal force.
Since the block is on a horizontal surface, the normal force \(N\) is equal to the weight of the block:
\[
N = mg
\]
Where:
- \(m = 5 \text{ kg}\) (mass of the block),
- \(g = 9.81 \text{ m/s}^2\) (acceleration due to gravity).
Calculating the normal force:
\[
N = 5 \times 9.81 = 49.05 \text{ N}
\]
Now calculating the maximum force of friction:
\[
f_f = 0.2 \times 49.05 = 9.81 \text{ N}
\]
Final Answer: The maximum force of friction acting on the block is 9.81 N.
Thermodynamics
Sample Problem 3: Heat Transfer
A metal block of mass 2 kg is heated from 20°C to 80°C. If the specific heat capacity of the metal is 500 J/(kg·°C), calculate the amount of heat energy absorbed by the block.
Solution:
The heat energy absorbed can be calculated using the formula:
\[
Q = mc\Delta T
\]
Where:
- \(Q\) is the heat energy,
- \(m = 2 \text{ kg}\) (mass),
- \(c = 500 \text{ J/(kg·°C)}\) (specific heat capacity),
- \(\Delta T = (80 - 20) = 60 \text{ °C}\) (temperature change).
Calculating the heat energy:
\[
Q = 2 \times 500 \times 60 = 60000 \text{ J}
\]
Final Answer: The amount of heat energy absorbed by the block is 60,000 J.
Sample Problem 4: Laws of Thermodynamics
A gas expands isothermally at a temperature of 300 K from a volume of 1 m³ to 3 m³. If the pressure of the gas is 100 kPa, calculate the work done by the gas during this expansion.
Solution:
For isothermal expansion, the work done can be calculated using the formula:
\[
W = P \Delta V
\]
Where:
- \(W\) is the work done,
- \(P = 100 \text{ kPa} = 100000 \text{ Pa}\) (pressure),
- \(\Delta V = V_f - V_i = 3 - 1 = 2 \text{ m}^3\) (change in volume).
Calculating the work done:
\[
W = 100000 \times 2 = 200000 \text{ J}
\]
Final Answer: The work done by the gas during the expansion is 200,000 J.
Electromagnetism
Sample Problem 5: Electric Circuits
A circuit consists of a 12 V battery connected to a resistor of 4 ohms. Calculate the current flowing through the circuit.
Solution:
Using Ohm's Law:
\[
V = IR
\]
Where:
- \(V = 12 \text{ V}\) (voltage),
- \(I\) is the current,
- \(R = 4 \text{ ohms}\) (resistance).
Rearranging the equation to solve for current:
\[
I = \frac{V}{R} = \frac{12}{4} = 3 \text{ A}
\]
Final Answer: The current flowing through the circuit is 3 A.
Sample Problem 6: Magnetic Fields
A long straight conductor carries a current of 10 A. Calculate the magnetic field intensity at a distance of 0.5 meters from the conductor.
Solution:
The magnetic field around a straight conductor can be calculated using the formula:
\[
B = \frac{\mu_0 I}{2\pi r}
\]
Where:
- \(B\) is the magnetic field intensity,
- \(\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A}\) (permeability of free space),
- \(I = 10 \text{ A}\) (current),
- \(r = 0.5 \text{ m}\) (distance from the conductor).
Substituting the values:
\[
B = \frac{(4\pi \times 10^{-7}) \times 10}{2\pi \times 0.5}
\]
\[
B = \frac{4 \times 10^{-6}}{1} = 4 \times 10^{-6} \text{ T} = 4 \mu T
\]
Final Answer: The magnetic field intensity at a distance of 0.5 meters from the conductor is \(4 \mu T\).
Waves and Oscillations
Sample Problem 7: Simple Harmonic Motion
A mass-spring system oscillates with a period of 2 seconds. What is the frequency of the oscillation?
Solution:
Frequency \(f\) is the reciprocal of the period \(T\):
\[
f = \frac{1}{T}
\]
Given that \(T = 2 \text{ s}\):
\[
f = \frac{1}{2} = 0.5 \text{ Hz}
\]
Final Answer: The frequency of the oscillation is 0.5 Hz.
Sample Problem 8: Wave Properties
A wave travels through a medium with a speed of 340 m/s and has a wavelength of 0.85 m. Calculate the frequency of the wave.
Solution:
The relationship between speed \(v\), frequency \(f\), and wavelength \(\lambda\) is given by:
\[
v = f\lambda
\]
Rearranging to solve for frequency:
\[
f = \frac{v}{\lambda} = \frac{340}{0.85} \approx 400 \text{ Hz}
\]
Final Answer: The frequency of the wave is approximately 400 Hz.
Conclusion
Physics sample problems with solutions are essential for students to solidify their understanding of fundamental concepts. By engaging with diverse problems across mechanics, thermodynamics, electromagnetism, and wave theory, learners can enhance their analytical and problem-solving skills. The systematic approach to solving these problems helps build a strong foundation for future studies in physics and related fields. Whether preparing for exams or seeking to deepen their understanding, students should regularly practice these types of problems to achieve greater proficiency in physics.
Frequently Asked Questions
What is the formula for calculating the force exerted by an object in motion?
The formula is F = ma, where F is the force, m is the mass of the object, and a is the acceleration.
How do you calculate the gravitational potential energy of an object?
Gravitational potential energy (U) is calculated using the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above the reference point.
What is the principle of conservation of energy?
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. The total energy in a closed system remains constant.
How can I determine the velocity of an object in free fall after 3 seconds?
The velocity (v) can be calculated using v = gt, where g is approximately 9.81 m/s² (acceleration due to gravity) and t is the time in seconds. So, v = 9.81 m/s² 3 s = 29.43 m/s.
What is the formula for calculating kinetic energy?
Kinetic energy (KE) is calculated using the formula KE = 1/2 mv², where m is the mass of the object and v is its velocity.
How do you solve for the acceleration of an object given initial and final velocities?
Acceleration (a) can be calculated using the formula a = (v_f - v_i) / t, where v_f is the final velocity, v_i is the initial velocity, and t is the time taken.
What is the formula for the period of a pendulum?
The period (T) of a simple pendulum can be calculated using the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.
How can I calculate the work done by a force?
Work (W) done by a force can be calculated using the formula W = Fd cos(θ), where F is the force, d is the distance moved in the direction of the force, and θ is the angle between the force and the direction of motion.
What is the relationship between pressure, force, and area?
Pressure (P) is defined as the force (F) applied per unit area (A). The formula is P = F/A, where P is pressure in pascals (Pa), F is force in newtons (N), and A is area in square meters (m²).
