exam 2 - practice exam 1

Instructions and Exam Strategy

This section outlines the expectations and rules for completing a physics exam. Students are required to answer all questions, show all work, and use the specified answer sheet. Academic honesty is emphasized, and the use of unauthorized aids is prohibited. Answers must be clear, boxed, and written legibly. Partial credit may be awarded for correct reasoning, even if the final answer is incorrect.

Useful Information and Constants

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

• Trigonometric identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $\sin(2\theta) = 2\sin\theta\cos\theta$

  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$

  • $\sin 45^\circ = \frac{1}{\sqrt{2}}$

• Earth's gravitational acceleration: $g = 9.8\ \mathrm{m/s^2}$ (toward Earth's center)

• Universal gravitational constant: $G = 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$

Gravity and Weightlessness

Weightlessness in Orbit

Objects in orbit, such as astronauts on the International Space Station, experience apparent weightlessness because they are in free fall around Earth. Both the station and its occupants are accelerating toward Earth at the same rate, resulting in no normal force acting on the astronauts.

• Key Point: Weightlessness is not due to the absence of gravity, but rather the continuous free-fall state.

• Example: An astronaut in orbit feels weightless because both she and the spacecraft are accelerating toward Earth at the same rate.

Newton's Law of Universal Gravitation

Gravitational Force Between Two Masses

Newton's law states that every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Formula: $F = G \frac{m_1 m_2}{r^2}$

• Doubling Masses: If both masses are doubled, the force increases by a factor of four.

• Example: If two balls A and B attract each other with force $F$, doubling both masses and the separation increases the force to $F$ (since $4F/4 = F$).

Work and Energy in Gravitational Fields

Work Done by Gravity

Gravity does work on objects as they move in its field. The sign of the work depends on the direction of motion relative to the force of gravity.

• Key Point: Work is positive when the object moves in the direction of gravity (downward), negative when moving against gravity (upward).

• Example: Throwing a baseball straight up: work by gravity is negative on the way up, positive on the way down.

Work in Circular Orbits

In a stable circular orbit, the gravitational force acts perpendicular to the direction of motion, so no work is done by gravity on the satellite.

• Key Point: No work is done by gravity in uniform circular motion.

Projectile Motion and Energy Conservation

Projectile Motion

When an object is thrown upward, it follows a parabolic trajectory under the influence of gravity. The maximum height is reached when the vertical velocity becomes zero.

• Formula for Maximum Height: $H = \frac{v_i^2}{2g}$ (for initial speed $v_i$ and no air resistance)

• Example: A pebble thrown upward with speed $v_i$ reaches height $H$; to reach $4H$, the required speed is $2v_i$.

Power Output

Power is the rate at which work is done. If one person does twice as much work in half the time as another, their power output is four times greater.

• Formula: $P = \frac{W}{t}$

Momentum and Collisions

Elastic and Inelastic Collisions

In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, only momentum is conserved.

• Key Point: When a ball bounces off a wall and reverses direction, its momentum changes, but kinetic energy may not be conserved if the collision is inelastic.

• Example: The momentum of the ball changes, but not necessarily its kinetic energy.

Impulse and Springs

Impulse and Spring Compression

Impulse is the change in momentum of an object, equal to the force applied times the time interval. When an object compresses a spring, the work done is stored as elastic potential energy.

• Formula for Spring Constant: $k = \frac{F}{x}$

• Elastic Potential Energy: $U = \frac{1}{2}kx^2$

• Impulse: $J = F \Delta t = \Delta p$

• Example: A cart compresses a spring by 1.25 m; the spring constant can be found using the work-energy principle.

Inclined Planes and Friction

Motion on an Incline

When an object moves up an incline, friction opposes the motion. The distance traveled before coming to rest can be found using energy conservation, accounting for work done against friction.

• Formula for Work Against Friction: $W_{fric} = \mu mg \cos\theta \cdot d$

• Energy Conservation: $\frac{1}{2}mv^2 = mgh + W_{fric}$

Circular Motion and Centripetal Force

Car on a Circular Hill

For a car to maintain contact with the road at the top of a circular hill, the normal force must be non-negative. The maximum speed is found by setting the normal force to zero and equating gravitational force to the required centripetal force.

• Formula: $mg = \frac{mv^2}{r}$

• Solving for $v$: $v_{max} = \sqrt{gr}$

Ballistic Pendulum and Conservation Laws

Ballistic Pendulum

A ballistic pendulum is used to measure the speed of a projectile. The collision is typically inelastic, so momentum is conserved during the collision, and energy is conserved during the subsequent swing.

• Momentum Conservation (collision): $m_{dart}v_{dart} = (m_{dart} + M_{block})v_{final}$

• Energy Conservation (swing): $\frac{1}{2}(m_{dart} + M_{block})v_{final}^2 = (m_{dart} + M_{block})gh$

• Mechanical Energy Loss: The difference in kinetic energy before and after the collision represents the energy lost, usually as heat or sound.

Summary Table: Conservation Laws in Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?
Elastic	Yes	Yes
Inelastic	Yes	No
Perfectly Inelastic (objects stick together)	Yes	No

Additional info:

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

• Specific problem numbers and diagrams referenced in the original file have been generalized for study purposes.