exam two - exan 2 practice test

Exam Instructions and Reference Information

General Guidelines

This section outlines the rules and procedures for completing the exam, emphasizing clarity, completeness, and academic honesty.

• Read all problems carefully and box your answers. Use the specified point value for each question.

• Show all work and reasoning to receive full credit. Partial credit may be awarded for incomplete solutions.

• Use the specified answer sheet for each question. Do not use scratch paper for final answers.

• Cross out all unattempted work to avoid losing points for incorrect statements.

• No notes, textbooks, or electronic devices are allowed during the exam.

• Reference card is provided for formulas and constants; no additional notes are permitted.

Reference Equations

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Trigonometric Identities: $\cos^2 \theta + \sin^2 \theta = 1$ $\cos 20^\circ$ $\sin 20^\circ$ $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\sin 0^\circ = 0$ $\sin 45^\circ = \frac{1}{\sqrt{2}}$

• Earth's Gravity: $g = 9.8\,\text{m/s}^2$ toward Earth's center

Mechanics: Forces, Energy, and Circular Motion

Weightlessness in Orbit

Understanding why astronauts feel weightless in orbit is fundamental to classical mechanics and gravitation.

• Key Point: Astronauts feel weightless because they are in free fall, continuously falling toward Earth but moving forward fast enough to stay in orbit.

• Example: An astronaut in a space station experiences apparent weightlessness due to the absence of a normal force acting on their body.

Friction and Circular Motion

Calculating the frictional force required to keep a car moving in a circle involves understanding centripetal force and friction.

• Key Point: The frictional force provides the necessary centripetal force for circular motion.

• Formula: $F_{\text{friction}} = \frac{mv^2}{r}$

• Example: For a car of mass $m = 3000\,\text{kg}$ moving at $30\,\text{m/s}$ on a curve of radius $r = 0.20\,\text{km}$, calculate $F_{\text{friction}}$.

Work, Energy, and Springs

Work and energy principles are applied to problems involving springs and stretching.

• Key Point: The work required to stretch a spring is given by $W = \frac{1}{2} k x^2$ where $k$ is the spring constant and $x$ is the displacement.

• Example: To stretch an ideal spring by $6.0\,\text{m}$ with $k = 417\,\text{N/m}$, $W = \frac{1}{2} \times 417 \times (6.0)^2$.

Tension in Circular Motion

Objects moving in a circle experience tension forces that can be calculated using Newton's laws.

• Key Point: Tension in the string provides the centripetal force for circular motion.

• Formula: $T = m \omega^2 r$

• Example: A stone of mass $1.0\,\text{kg}$ tied to a string and swung in a circle of radius $0.80\,\text{m}$ at $40\,\text{rad/s}$: $T = 1.0 \times (40)^2 \times 0.80$.

Work and Power in Climbing

Calculating the minimum rate of work (power) required to climb stairs involves energy and time.

• Key Point: Power is the rate at which work is done: $P = \frac{W}{t}$

• Formula: $W = mgh$

• Example: A boy of mass $60.0\,\text{kg}$ climbs $10.0\,\text{m}$ in $8.0\,\text{s}$: $P = \frac{60.0 \times 9.8 \times 10.0}{8.0}$

Collision and Conservation of Momentum

Collisions are analyzed using conservation of momentum and energy principles.

• Key Point: In an inelastic collision, total momentum is conserved but kinetic energy is not.

• Formula: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$

• Example: Two blocks of masses $5\,\text{kg}$ and $7\,\text{kg}$ collide and stick together; find final velocity.

Gravitational Force and Orbits

Gravitational Force Calculation

Newton's law of universal gravitation is used to calculate the force between two masses.

• Key Point: The gravitational force between two masses is $F = G \frac{m_1 m_2}{r^2}$

• Example: Calculate the force between the Earth and a space station at a given distance.

Radial Acceleration in Orbit

Objects in circular orbit experience radial (centripetal) acceleration.

• Key Point: Radial acceleration is $a_r = \frac{v^2}{r}$

• Example: For a space station orbiting at radius $r$, calculate $a_r$ in terms of $g$.

Geosynchronous Orbit

Geosynchronous orbits are those where the orbital period matches Earth's rotation period.

• Key Point: The radius for a geosynchronous orbit can be found by equating gravitational force to centripetal force and setting $T = 24$ hours.

• Formula: $\frac{G M}{r^2} = \frac{4 \pi^2 r}{T^2}$

Loop-the-Loop and Energy Conservation

Energy at the Top of a Loop

Analyzing a car moving through a vertical loop involves conservation of energy and forces.

• Key Point: At the top of the loop, the car must have enough speed to maintain contact with the track.

• Formula: $KE_{\text{top}} = mgh$

• Normal Force: $N = m \frac{v^2}{R} - mg$

• Example: Calculate the minimum speed and normal force for a car at the top of a loop of radius $R$.

Inclined Plane and Friction

Work Against Friction on an Incline

Calculating how high a block travels up an incline with friction involves energy conservation and work done against friction.

• Key Point: The work done against friction is $W_f = F_f \times d$

• Formula: $PE = KE - W_f$

• Example: Two blocks connected by a string move up an incline; calculate the vertical height reached.

Summary Table: Key Formulas and Concepts

Concept	Formula	Application
Gravitational Force	$F = G \frac{m_1 m_2}{r^2}$	Force between Earth and satellite
Radial Acceleration	$a_r = \frac{v^2}{r}$	Acceleration in circular motion
Work (Spring)	$W = \frac{1}{2} k x^2$	Stretching a spring
Conservation of Energy	$PE + KE = \text{constant}$	Loop-the-loop, inclined plane
Power	$P = \frac{W}{t}$	Climbing stairs
Momentum Conservation	$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$	Collisions

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Where original questions or solutions were fragmented, standard physics principles and formulas have been logically inferred and included.