Physics 407: Practice Final Exam Study Notes

Vectors and Vector Operations

Graphical Addition and Subtraction of Vectors

Vectors are quantities that have both magnitude and direction. Understanding how to add and subtract vectors graphically is fundamental in physics, especially in kinematics and dynamics.

• Graphical Addition: To add vectors A and B, place the tail of B at the head of A. The resultant vector A + B is drawn from the tail of A to the head of B.

• Graphical Subtraction: To subtract vector B from A, reverse the direction of B to get -B, then add it to A using the graphical method above.

• Example: If A points right and B points up, A + 2B is found by doubling B and adding it to A as described.

Force and Motion

Free-Body Diagrams

Free-body diagrams are essential tools for analyzing the forces acting on an object. Each force is represented as an arrow pointing in the direction the force acts.

• Key Forces: Weight (gravity), normal force, tension, friction, and applied forces.

• Example: For a car at the top of a hill, the forces include gravity (downward) and the normal force (upward from the road).

Normal Force on a Hill

When a car moves over the top of a hill of radius r at speed v, the normal force changes due to the requirement for centripetal acceleration.

• Equation: At the top of the hill, the sum of forces provides the centripetal force: $N + (-mg) = -m\frac{v^2}{r}$ where N is the normal force, m is mass, g is acceleration due to gravity, and v is speed.

• Critical Speed: The car loses contact when N = 0, so: $mg = m\frac{v_{\text{crit}}^2}{r}$ $\Rightarrow v_{\text{crit}} = \sqrt{gr}$

Dynamics: Newton's Laws and Applications

Elevator Free-Body Diagram

When an elevator slows down, the forces acting on it include gravity and the tension in the cable. The direction and magnitude of these forces determine the acceleration.

• Newton's Second Law: $\sum F = ma$

• Example: If the elevator is slowing down while moving downward, the tension must be greater than the weight.

Impulse and Momentum: Bullet and Block Collision

Collisions involve the conservation of momentum. When a bullet embeds in a block, the combined system moves with a new velocity.

• Conservation of Momentum: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$ where $m_1$ and $v_1$ are the mass and velocity of the bullet, $m_2$ and $v_2$ are those of the block, and $v_f$ is the final velocity.

• Penetration Depth: The distance the bullet travels in the block can be found using work-energy principles.

Forces in Ropes and Tension

Maximum Tension in a Rope

The maximum force a rope can withstand before breaking is its tensile strength. When two people pull on a rope from opposite ends, the tension is equal to the force each applies (assuming the rope is massless and there is no acceleration).

• Key Point: The tension does not double; it remains equal to the force applied by each person.

• Looped Rope: If the rope is looped and pulled, the tension at the weakest point is still determined by the force applied, not doubled.

Newton's Third Law and Free-Body Diagrams

Car and Truck Interaction

When two objects interact, the force exerted by object A on object B is equal in magnitude and opposite in direction to the force exerted by B on A.

• Newton's Third Law: $F_{A \to B} = -F_{B \to A}$

• Example: A car pushing a truck: draw separate free-body diagrams for each, showing all forces including the interaction force between them.

Dynamics II: Circular Motion and Banked Curves

Banked Curves Without Friction

Banked curves allow vehicles to turn without relying on higher friction. The banking angle provides the necessary centripetal force for circular motion.

• Free-Body Diagram: Forces include gravity and the normal force, which has components both perpendicular and parallel to the surface.

• Banking Angle Equation: $\tan \theta = \frac{v^2}{gR}$ where $\theta$ is the banking angle, $v$ is speed, $g$ is gravity, and $R$ is the radius of the curve.

Potential Energy and Equilibrium

Potential Energy Diagrams

The potential energy function $U(x)$ describes the energy stored in a system as a function of position. The force is related to the negative gradient of the potential energy.

• Force from Potential: $F(x) = -\frac{dU}{dx}$

• Equilibrium Points: Points where $\frac{dU}{dx} = 0$. Stable equilibrium occurs at minima, unstable at maxima.

• Example: For a particle in a potential well, estimate the force at given positions and identify equilibrium points from the graph.

Energy Conservation and Minimum Speed

To find the minimum speed required for a particle to reach a certain position, use conservation of energy:

• Equation: $\frac{1}{2}mv_0^2 + U(x_0) = \frac{1}{2}mv^2 + U(x)$

• Application: Set final kinetic energy to zero to solve for the minimum initial speed.

Newton's Theory of Gravity

Gravitational Attraction Between Two Stars

Newton's law of universal gravitation describes the force between two masses:

• Equation: $F = G\frac{M_1 M_2}{r^2}$ where $G$ is the gravitational constant, $M_1$ and $M_2$ are the masses, and $r$ is the distance between centers.

• Energy Conservation: The change in gravitational potential energy equals the gain in kinetic energy as the stars move toward each other.

• Speed Before Impact: Use conservation of energy: $\Delta U = \Delta K$ $-G\frac{M_1 M_2}{r_{\text{final}}} + G\frac{M_1 M_2}{r_{\text{initial}}} = \frac{1}{2}M_1 v_1^2 + \frac{1}{2}M_2 v_2^2$