1.

A force is said to be conservative if the work

done by the force on an object is independent
of the path taken by the object. This means
that the work done by the force on an object
only depends on the starting and ending
positions of the object, and not on the path
taken between them. Conservative forces are
directly related to potential energy, as the work
done by a conservative force can be expressed
as a change in potential energy. When an
object moves in the direction of a conservative
force, the potential energy decreases, and
when it moves in the opposite direction, the
potential energy increases.

2. Heat transfer is the process of energy

exchange between two systems due to a
temperature difference. It occurs across the
boundary of a system when the system is not
in a state of thermal equilibrium. There are
three modes of heat transfer: conduction,
convection, and radiation. The effects of heat
transfer on a matter include changes in
temperature, phase changes, and changes in
physical and chemical properties.

3. The thermodynamic processes of a petrol

and diesel engine can be represented using
different PV diagrams. In a petrol engine, the
fuel and air mixture is compressed in the
cylinder, then ignited by a spark. This causes
an increase in pressure, which pushes the
piston down, releasing energy. In a diesel
engine, the air is compressed first, then fuel is
injected into the cylinder. The fuel ignites due
to the high temperature and pressure, causing
an increase in pressure and releasing energy.
The difference lies in the way the fuel
is ignited, which affects the thermodynamic
properties of the system.

4. The three charging processes are constant

current charging, constant voltage charging,
and trickle charging. Constant current charging
involves charging the battery with a constant
current until it reaches a desired state of
charge. Constant voltage charging involves
charging the battery with a constant voltage
until it reaches a desired state of charge.
Trickle charging is a slow charging process that
maintains the battery's charge at a lower
voltage.

5. The potential energy of the cart at point A

relative to the ground can be calculated using
mgh1, where m is the mass of the cart, g is the
acceleration due to gravity, and h1 is the height
of point A. The potential energy is
equal to 300 kg x 9.81 m/s² x 25 m = 73,575 J.
Neglecting friction, the speed of the cart at
point B can be found by using the conservation
of energy. The initial potential energy at point A
is converted into kinetic energy at point B. The
equation for this is mgh1 = 1/2mv² + mgh2.
Solving for v gives v = sqrt(2gh1 - 2gh2) = 30.5
m/s.

6. The total work done by the frictional force is

equal to the negative of the total change in
kinetic energy of the block. Since there is no
change in potential energy due to the
horizontal slide, this means that the work done
by the frictional force is equal to the initial
potential energy minus the final kinetic energy.
The initial potential energy is mgh, where m is
the mass of the block, g is the acceleration due
to
gravity, and h is the initial height. The final
kinetic energy can be found using 1/2mv²,
where v is the final velocity. Solving for v using
conservation of energy gives v = sqrt(2gh -
2uxk/m) = 3.14 m/s. The distance the block
slides along the horizontal surface before
coming to rest can be found using the equation
x = v²/(2g(u + (sin θ) - cos θ))) = 0.51 m.

7. (a) The period of oscillation of the block can

be found using the equation T = 2π/ω, where T
is the period, and ω is the angular frequency.
The angular frequency can be found using ω =
sqrt(k/m), where k is the spring constant, and
m is the mass of the block. Thus, T = 2π
sqrt(m/k) = 2π sqrt((0.25*4)/4) = π s. (b) At the
time t = (2/3)T, the block is at a position x =
(2/3)A. Thus, the potential energy at this time is
mgh = mg(0.15 + 0.25)sin(35º) = 1.475 J. The
kinetic energy at this time can be found using
1/2mv², where v is the velocity at this point.
Using conservation of energy, the total
mechanical energy must be equal to the sum
of the kinetic and potential energies. Thus,
1/2mv² + mgh = mgh1, where h1 is the initial
height.
Solving for v gives v = sqrt(2(gh1 - h - uxk/m))
= 0.65 m/s.

8. The acceleration of the mass can be found

by taking the second derivative of the
displacement equation, which gives a = -ω²A
sin(ωt). The angular frequency can be found
using ω = 2πf, where f is the frequency in Hz.
Converting 1050 vibrations per minute to Hz
gives f = 17.5 Hz. Thus, ω = 110 π rad/s. At the
mid-distance from the equilibrium position, the
displacement is A/2. Thus, the acceleration at
this point is a = -ω²(A/2) = -605 m/s². The
speed at this point can be
found using the equation for simple harmonic
motion, v = ωAcos(ωt). At the mid-distance, the
cosine of the angle is 0, so the speed at this
point is zero.
9. The major causes and differences of the four
wave behaviors are:

- Reflection: Occurs when a wave encounters

a boundary and bounces back into the same
medium. The angle of incidence equals the
angle of reflection.
- Refraction: Happens when a wave enters a
new medium at an angle, causing its speed to
change and its direction to bend. This bending
is due to the change in the wave's speed as it
crosses the boundary between two different
mediums.
- Interference: Results from the superposition
of two or more waves, either constructive
(waves adding up to create a wave with greater
amplitude) or destructive (waves canceling
each other out).
- Diffraction: Occurs when waves encounter
an obstacle or aperture, causing them to bend
around it and spread out. The extent of
diffraction depends on the size of the obstacle
or aperture relative to the wavelength of the
wave.

10. To solve for the new length of the brass rod

when its temperature changes, you can use
the formula for linear thermal expansion:

\[
 L_f = L_i \times (1 + \alpha \times \Delta T)
\]

where \(L_f\) is the final length, \(L_i\) is the

initial length, \(\alpha\) is the coefficient of
linear expansion, and \(\Delta T\) is the change
in temperature.

11. To calculate the amount of heat needed to

transfer the ice from -100°C to steam at
1200°C, you need to consider the heat
required for each phase change (melting,
heating, boiling) using the specific heat
capacities of ice, water, and steam, as well as
the latent heats of fusion and vaporization.

12.
a. To find the electric field at a point on the y-
axis due to the two point charges, you can use
the formula for electric field due to a point
charge:
\[E = \frac{k \cdot |q|}{r²} \]
where \(k\) is Coulomb's constant, \(q\) is the
charge, and \(r\) is the distance from the
charge to the point. Then, use vector addition
to find the net electric field.
 b. Use Coulomb's law to calculate the
electric force on the -3.00μC charge placed at
the given position due to each of the two point
charges, then sum the forces to find the total
electric force.

13. a. The equivalent resistance of the network

can be found
using the formulas for resistors in series and
parallel configurations.
b. Use Ohm's law to find the total current
flowing through the resistors, knowing the
equivalent resistance and the source voltage.

14. To find the linear speed of the charged

body moving in a circular orbit in a magnetic
field, you can use the formula for centripetal
force and equate it to the magnetic force
experienced by the charged body due to its
circular motion in the magnetic field.

15. a. When a moving charged body passes

nearby a bar magnet, it changes its path due to
the Lorentz force, which acts perpendicular to
both the velocity of the charged body and the
magnetic field.
b. Two current-carrying wires repel or attract
each other due to the magnetic force between
them, which arises from the interaction of their
magnetic fields.

16. To determine the initial speed required for

the object to reach the top of the circle, you
can use the conservation of mechanical
energy, considering the gravitational potential
energy at the bottom of the circle and the
kinetic energy at the top of the circle.

Certainly! Here is the formula written in a

formal format:
The electric field (E) generated by a point
charge with
magnitude |q| can be calculated using the
equation:

9. The major causes and differences of the four

wave behaviors are:

- Reflection: Occurs when a wave encounters

a boundary and bounces back into the same
medium. The angle of incidence equals the
angle of reflection.
- Refraction: Happens when a wave enters a
new medium at an angle, causing its speed to
change and its direction to bend. This bending
is due to the change in the wave's speed as it
crosses the boundary between two different
mediums.
- Interference: Results from the superposition
of two or more waves, either constructive
(waves adding up to create a wave with greater
amplitude) or destructive (waves canceling
each other out).
- Diffraction: Occurs when waves encounter
an obstacle or aperture, causing them to bend
around it and spread out. The extent of
diffraction depends on the size of the obstacle
or aperture relative to the wavelength of the
wave.

10. To solve for the new length of the brass rod

when its temperature changes, you can use
the formula for linear thermal expansion:

Lf = Li \times (1 + \alpha \times \Delta T)

 where (Lf) is the final length, (Li) is the initial
length,
(alpha) is the coefficient of linear expansion,
and (Delta T) is the change in temperature.

11. To calculate the amount of heat needed to

transfer the ice from -100°C to steam at
1200°C, you need to consider the heat
required for each phase change
(melting, heating,boiling) using the specific
heat capacities of ice, water, and steam, as
well as the latent heats of fusion and
vaporization.

12. a. To find the electric field at a point on the

y-axis due to the two point charges, you can
use the formula for electric field due to a point
charge:
[ E = frac{k \cdot |q|}{r²} \]
where (k) is Coulomb's constant, (q) is the
charge, and (r) is the distance from the charge
to the point. Then, use vector addition to find
the net electric field.
b. Use Coulomb's law to calculate the
electric force on the -3.00μC charge placed at
the given position due to each of the two point
charges, then sum the forces to find the total
electric force.

13. a. The equivalent resistance of the

network can be found using the formulas for
resistors in series and parallel configurations.
b. Use Ohm's law to find the total current
flowing through
the resistors, knowing the equivalent
resistance and the source voltage.

14. To find the linear speed of the charged

body moving in a circular orbit in a magnetic
field, you can use the formula for centripetal
force and equate it to the magnetic force
experienced by the charged body due to its
circular motion in the magnetic field.

15. a. When a moving charged body passes

nearby a bar magnet, it changes its path due to
the Lorentz force, which acts perpendicular to
both the velocity of the charged body and the
magnetic field.
b. Two current-carrying wires repel or attract
each other due to the magnetic force between
them, which arises from the interaction of their
magnetic fields.

16. To determine the initial speed required for

the object to reach the top of the circle, you
can use the conservation of mechanical
energy, considering the gravitational potential
energy at the bottom of the circle and the
kinetic energy at the top of the circle.
[E = \frac{k \cdot |q|}{r²}]
where:
- E is the electric field strength,
- k is Coulomb's constant (approximately 9 x
10^9 Nm²/C²),
- |q| is the magnitude of the point charge, and
- r is the distance from the point charge at
which you want to
evaluate the electric field.
Feel free to let me know if there's anything else
I can assist you with!
17. To find the work done, integrate the force
function from the initial position \(x = 0\) to the
final position (x = 8):

W = (int{0}^{8} (3x² - 4x + 5) \, dx\]

18.
a. To find the speed of the race car, you can
use the relationship between the frequency of
the sound emitted by the engine and the
number of revolutions it makes per kilometer.
b. Convert the revolutions per kilometer to
revolutions per minute to determine the
engine's rotation speed.

19.
a. Use the formula for maximum velocity in
simple harmonic motion to find the maximum
velocity of the person bouncing on the
bathroom scale.
b. Calculate the maximum energy stored in
the spring using the formula for the potential
energy of a spring.

20.
a. Plug in \(t = 1.5\) sec into the given
equation to find the displacement \(x\).
 b. To find the speed, differentiate the given
equation with respect to time and then plug in \
(t = 1.5\) sec.
c. Similarly, differentiate the velocity
equation to find acceleration at \(t = 1.5\) sec.

21.
a. The amplitude of harmonic oscillations
can be determined from the given initial
velocity and spring constant.
b. The maximal acceleration occurs at
maximum displacement from equilibrium
position.
c. Use the equation of motion for simple
harmonic motion to determine the full time
dependence \(x(t)\).
d. To find the time to achieve maximal
displacement for the first time, consider the
initial conditions and the equation of motion.

22. To calculate the ratio of heat flow in the two

rods, you can use Fourier's Law of heat
conduction, which states that the rate of heat
transfer is proportional to the temperature
gradient. Then, consider the temperatures at
the ends of the rods and their thermal
conductivities to find the ratio of heat flow.
OR
17. To find the work done, you
need to
integrate the force over the
distance traveled:

\[ W = \int_{x_1}^{x_2} F(x) \,
dx \]

Substituting the given force

function, integrate from \(x = 0\)
to \(x = 8\) to find the work done.

18. a. To find the speed of the

race car, you can use the formula
for the frequency of sound waves
emitted by a moving source:

\[ f' = f \left( \frac{v}{v - v_s} \

right) \]
 where \(f\) is the frequency
of the sound, \(v\) is the speed of
sound in air, and \(v_s\) is the
speed of the source.

b. Given the number of

revolutions per
kilometer, you can calculate the
revolutions per minute by
converting kilometers per minute
to revolutions per minute.

19. a. To find the maximum

velocity of the person bouncing
on the bathroom scale, you can
use the relationship between
maximum velocity and amplitude
in simple harmonic motion.
b. The maximum energy
stored in the spring can be
calculated using the formula for
elastic potential energy in a
spring system.

20. Given the equation of motion

for simple harmonic motion
(SHM), you can directly calculate
the displacement, velocity, and
acceleration of the body at
(t = 1.5ş)seconds by substituting
the given
time into the equation.

21. a. The amplitude of the

harmonic oscillations is the
maximum displacement from the
equilibrium position, which can
be found directly from the
equation of motion.
b. The maximal acceleration
occurs at the amplitude of the
oscillation, and it can be
calculated using the formula for
acceleration in SHM.
c. The full time dependence
(x(t)) can be obtained by
substituting the given equation of
motion into the equation for
SHM.
 d. To find the time to achieve
maximal displacement for the
first time, you can analyze the
motion at the amplitude using
the equation of motion.

22. To calculate the ratio of heat

flow in the two rods, you can use
Fourier's law of heat conduction,
which states that the rate of heat
transfer through a material is
directly proportional to the
temperature gradient. Evaluate
the heat flow in each rod and
then calculate the ratio.
23. To find the speed of the
particle when it is at a distance of
10m from the fixed charge, you
can use the principle of
conservation of mechanical
energy. Initially, the particle has
gravitational potential energy and
electric potential energy, and as
it moves towards the fixed
charge, this energy is converted
into kinetic energy.

The total mechanical energy at

the initial position (Ei) is the sum
of gravitational
potential energy (mgh)and
electric potential energy ((k\
frac{q1q2}{ri})), where (m) is the
mass of the particle,
(g) is the acceleration due to
gravity,
(h) is the height,
(k) is Coulomb's constant,
(q1) and (q2) are the charges,
and
(ri) is the initial distance.

Similarly, the total mechanical

energy at the final position (\
(E_f\)) is the kinetic energy (\(\
frac{1}{2}mv^2\)) plus the electric
potential energy at the final
distance (\(k\frac{q_1q_2}{r_f}\)),
where \(v\) is the final velocity
and \(r_f\) is the final distance.

Since mechanical energy is

conserved
(Ei = Ef), you can set up the
equation and solve for (v).
[ mgh + k frac{q_1q_2}{r_i} = \
frac{1}{2}
mv^2 + k\frac{q_1q_2}{r_f} \]

24. To calculate the potential

difference between points X and
Y in the network,
you can sum up the potential
differences across each
component
(resistors and batteries) along
the path connecting X to Y.
Use Ohm's law for resistors (\(V
= IR\)) and the formula for the
potential difference of a battery
(\(V = \epsilon - Ir\)). Then, sum
up the potential differences along
the path to find \(V_{XY}\).