A block of mass $5$ kg is placed on a frictionless inclined plane making an angle of $30$ with the horizontal. Calculate the acceleration of the block and the time it takes to slide $2$ meters down the plane from rest.

Electromagnetism:

A circular loop of radius $0\mathrm{.}1$ meters carries a current of $5$ A$.$ Calculate the magnetic field at the center of the loop.

Thermodynamics:

A Carnot engine operates between two heat reservoirs at temperatures of $500$ K and $300$ K$.$ If the engine absorbs $600$ J of heat from the hot reservoir, calculate the work done by the engine and the heat rejected to the cold reservoir.

Optics:

A double$-$slit experiment is performed with light of wavelength $600$ nm$.$ The slits are separated by $0\mathrm{.}5$ mm and the screen is $2$ meters away from the slits. Calculate the distance between the central maximum and the first$-$order maximum on the screen.

Quantum Mechanics:

Calculate the energy of the first excited state of an electron in a one$-$dimensional infinite potential well of width $0\mathrm{.}1$ nm$.$ Use the particle in a box model.

Electric Circuits:

A series RLC circuit has a resistance of $10$ ohms, an inductance of $0\mathrm{.}1$ H$,$ and a capacitance of

$10$

$$

$$

$10$F$.$ Calculate the resonant frequency of the circuit.

Relativity:

Calculate the relativistic kinetic energy of an electron moving at $0\mathrm{.}8$c$,$ where

$$

c is the speed of light.

Fluid Mechanics:

Water flows through a pipe of diameter $0\mathrm{.}2$ meters with a velocity of $2$ m$/$s$.$ Calculate the Reynolds number and determine whether the flow is laminar or turbulent. Assume the kinematic viscosity of water is

$1$

$$

$1$

$0$

$$

$6$

$$

$$

$2$

$/$

$$

$110$

$6$

m

$2$

$/$s$.$

Astrophysics:

Calculate the Schwarzschild radius of a black hole with a mass $10$ times that of the Sun. Use the formula

$$

$$

$=$

$2$

$$

$$

$$

$2$

R

s

$=$

c

$2$

$2$GM

$,$ where

$$

G is the gravitational constant,

$$

M is the mass, and

$$

c is the speed of light.

Nuclear Physics:

A sample contains $1$ gram of

$238$

$$

$238$

U$.$ Calculate the number of

$238$

$$

$238$

U atoms in the sample and the activity $($decay rate$)$ if the half$-$life of

$238$

$$

$238$

U is

$4\mathrm{.}468$

$$

$1$

$0$

$9$

$4\mathrm{.}46810$

$9$

years.

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