Physics 211: Forces, Motion, Work, Energy, and Momentum – Midterm 2 Review Study Notes

Forces and Motion

Newton's Laws of Motion

Newton's laws describe the relationship between the motion of an object and the forces acting on it. They are foundational to classical mechanics.

• Newton's First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

$F_{net} = ma$

• Newton's Third Law: For every action, there is an equal and opposite reaction. Forces always occur in pairs between two different objects.

Example: In a free-body diagram, action-reaction pairs do not appear because they act on different objects.

Friction

Static and Kinetic Friction

Friction is a force that opposes the relative motion or tendency of such motion of two surfaces in contact.

• Static friction ($f_s$): The frictional force that prevents relative motion up to a maximum value.

• Kinetic friction ($f_k$): The frictional force acting when two surfaces are sliding past each other.

$f_s \leq \mu_s N$ $f_k = \mu_k N$

Example: To keep a box of mass 18 kg from sliding down a wall, the minimum force $F$ required is found by balancing forces and using the static friction coefficient ($\mu_s = 0.9$):

$F = \frac{mg}{\mu_s} = \frac{18 \times 9.8}{0.9} = 200\ \text{N}$

Work and Kinetic Energy

Work Done by a Force

Work is done when a force causes displacement. The work done by a constant force is:

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• If the force is perpendicular to the displacement, no work is done (e.g., carrying an object at constant speed across a flat surface).

Kinetic Energy and the Work-Energy Theorem

Kinetic energy ($K$) is the energy of motion:

$K = \frac{1}{2}mv^2$

The work-energy theorem states that the net work done on an object equals its change in kinetic energy:

$W_{net} = \Delta K = K_f - K_i$

Potential Energy

Gravitational Potential Energy

Potential energy is energy stored due to an object's position. For gravity near Earth's surface:

$U_g = mgh$

Mechanical energy is conserved if only conservative forces (like gravity) do work:

$E = K + U$

Example: For a rollercoaster car starting from rest at height $h$, the speed at a lower height is found using energy conservation:

$mgh = \frac{1}{2}mv^2 \implies v = \sqrt{2gh}$

Momentum and Impulse

Linear Momentum

Momentum ($p$) is the product of mass and velocity:

$p = mv$

Impulse is the change in momentum caused by a force acting over a time interval:

$J = F\Delta t = \Delta p$

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant.

Example: If a person on a cart throws a ball to the right, the cart moves to the left to conserve momentum.

Applications and Problem-Solving Strategies

Free-Body Diagrams

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

Grading of Exam Problems

• Define all relevant quantities (often with a figure).

• Explain which physics principles apply (e.g., energy conservation, Newton's laws).

• Set up the equations to be solved.

• Solve the equations (the calculation is less important than the setup and reasoning).

• No points are given for the result alone; reasoning and method are most important.

Worked Examples

Work and Forces

• Carrying an object at constant speed: No work is done by the gravitational force if it is perpendicular to the velocity.

Newton's Third Law in Free-Body Diagrams

• Action-reaction pairs do not appear in a single object's free-body diagram; they act on different objects.

Normal Force Adjustment

• When an upward force is applied to an object on a table, the normal force decreases as the applied force increases, keeping the sum of vertical forces zero until the object lifts off.

Momentum Conservation Example

• When a ball is thrown and bounces off a barrier on a cart, the cart moves in the opposite direction to conserve momentum.

Friction and Forces Example

• To keep a box at rest against a wall, the minimum force is determined by balancing gravity and friction, using the static friction coefficient.

Energy Conservation Example

• For a rollercoaster, use conservation of mechanical energy to find speed at different points:

$mgh + \frac{1}{2}mv_1^2 = \frac{1}{2}mv_2^2$

Work-Energy and Stopping Distance

• To find stopping distance when a car skids to a stop:

$W = K_f - K_i = -\mu_k mg \Delta x = -\frac{1}{2}mv^2$ $\Delta x = \frac{v^2}{2\mu_k g}$

• The stopping distance does not depend on mass.

• All initial kinetic energy is transformed into heat due to friction.

Collisions and Energy Loss

• When colliding with another car or a truck, the fraction of kinetic energy lost depends on the masses involved.

Scenario	Fraction of $K_i$ Lost	Explanation
Car-car ($m_1 = m_2$)	0.5	Best option; 50% lost
Car-truck ($m_2 = 9m_1$)	0.9	90% lost
Car-heavy truck ($m_2 \gg m_1$)	1.0	100% lost

Example: Colliding with a car of equal mass loses less kinetic energy than colliding with a much heavier truck.

Summary Table: Key Equations

Concept	Equation
Newton's Second Law	$F_{net} = ma$
Friction (static/kinetic)	$f_s \leq \mu_s N$, $f_k = \mu_k N$
Work	$W = Fd\cos\theta$
Kinetic Energy	$K = \frac{1}{2}mv^2$
Potential Energy (gravity)	$U_g = mgh$
Momentum	$p = mv$
Impulse	$J = F\Delta t = \Delta p$
Stopping Distance	$\Delta x = \frac{v^2}{2\mu_k g}$

Additional info: These notes are based on a midterm review and include both conceptual and quantitative problems, with emphasis on reasoning, setup, and application of fundamental physics principles.