## Learning Outcomes

- Describe translational motion using displacement, velocity, acceleration, vectors, and scalars.
- Apply Newton’s laws to forces including weight, normal force, tension, friction, and drag.
- Calculate work, kinetic energy, gravitational potential energy, elastic potential energy, and power.
- Use the work–energy theorem and conservation of mechanical energy to solve MCAT-style problems.
- Distinguish conservative from nonconservative forces and predict how each affects mechanical energy.
- Analyze translational and rotational equilibrium using net force and net torque conditions.
- Explain mechanical advantage in inclined planes, levers, and pulleys, especially in biological settings.

## Medical College Admission Test (MCAT) Syllabus

In the Chemical and Physical Foundations of Biological Systems section, mechanics appears within Foundational Concept 4, which tests how physical principles explain biological systems. The section contains 59 questions in 95 minutes, with both passage-based and discrete questions. Physics makes up about one quarter of the section, and Foundational Concept 4 accounts for a large share of the Chemical/Physical content. For 4A, expect calculations, graph interpretation, unit analysis, and biological applications such as muscle work, joint forces, blood vessel drag, and movement of objects or body parts.

- Translational motion: displacement, speed, velocity, acceleration, and constant-acceleration relationships
- Vectors and scalars: components, direction, magnitude, and sign conventions
- Forces: Newton’s laws, weight, normal force, tension, friction, air resistance, and viscous drag
- Work and energy: work as energy transfer, kinetic energy, potential energy, power, and efficiency
- Conservation principles: mechanical energy conservation and energy dissipation by nonconservative forces
- Equilibrium: net force, net torque, center of mass, and stable mechanical balance
- Simple machines: inclined planes, levers, pulleys, mechanical advantage, and biological lever systems

## Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision.

1. A 2 kg object moving at 3 m/s doubles its speed. By what factor does its kinetic energy change?
2. Why can an object be in equilibrium even if it is moving?
3. A force of 20 N is applied at 60° to a displacement of 5 m. How much work is done by the force?
4. What two conditions must be true for a rigid body to be in static equilibrium?
5. Why does a longer ramp reduce the force needed to raise a patient on a wheelchair, even though the work output is similar?

## Introduction

Mechanics is the study of motion and the forces that cause motion. On the MCAT, this topic is rarely pure plug-and-chug physics. Instead, passages often connect forces, energy, and equilibrium to living systems: a muscle contracting, a bone acting as a lever, a fluid resisting motion, or a person climbing stairs. Your task is to translate the situation into a physical model, choose the correct equation, and keep track of signs, directions, and units.

The most useful habit for 4A is to ask: “Is this a force problem, an energy problem, or an equilibrium problem?” Force problems usually require free-body diagrams and Newton’s laws. Energy problems often use work, kinetic energy, and potential energy. Equilibrium problems require balanced forces and, when rotation is possible, balanced torques.

> **Key Term: translational motion**
> Motion in which an object changes position without requiring analysis of rotation about an axis.

### Describing Motion with Vectors, Velocity, and Acceleration

Translational motion begins with position. If an object moves from one location to another, its displacement is the change in position, including direction. Displacement is a vector; distance is a scalar. For example, if a patient walks 10 m east and then 10 m west, the total distance is 20 m, but the displacement is 0 m.

> **Key Term: displacement**
> The change in position of an object, including both magnitude and direction.

Velocity is the rate of change of displacement, while speed is the magnitude of velocity. A car moving north at 20 m/s and a car moving south at 20 m/s have the same speed but different velocities. Kinetic energy depends on speed, not velocity direction, so those two cars have the same kinetic energy if their masses are equal.

Acceleration is the rate of change of velocity. An object accelerates if it speeds up, slows down, or changes direction. For straight-line motion with constant acceleration, the main relationships are:

$$
v_f = v_i + at
\Delta x = v_i t + \frac{1}{2}at^2
v_f^2 = v_i^2 + 2a\Delta x
$$

where initial velocity, final velocity, acceleration, time, and displacement must all use consistent signs. Near Earth’s surface, freely falling objects have acceleration due to gravity of about 9.8 m/s² downward, commonly approximated as 10 m/s² for quick MCAT arithmetic.

> **Test Tip:**
> If time is not given and not asked for, try the equation with velocities, acceleration, and displacement: $$v_f^2 = v_i^2 + 2a\Delta x$$. If displacement is not involved, use $$v_f = v_i + at$$.

Graphs are common. The slope of a position–time graph gives velocity. The slope of a velocity–time graph gives acceleration. The area under a velocity–time graph gives displacement. The area under an acceleration–time graph gives change in velocity.

> **Key Term: velocity**
> The rate of change of displacement; it is a vector with magnitude and direction.

MCAT questions may use biological motion: a red blood cell traveling through a capillary, a runner accelerating from rest, or a lifted object slowing at the top of its path. In each case, define a positive direction before assigning signs.

### Forces, Newton’s Laws, and Free-Body Diagrams

A force is an interaction that can cause acceleration. Newton’s second law is the key equation:

$$
\sum F = ma
$$

The left side is net force, not just any force in the problem. If multiple forces act, add them as vectors. If the net force is zero, acceleration is zero.

> **Key Term: force**
> A vector interaction that can change an object’s motion; the SI unit is the newton, equal to kg·m/s².

Newton’s first law states that an object remains at rest or in constant-velocity motion unless acted on by a net external force. This is why equilibrium does not always mean “stationary.” An object moving with constant velocity has zero acceleration and zero net force.

Newton’s third law states that forces occur in equal and opposite pairs acting on different objects. If the foot pushes backward on the ground, the ground pushes forward on the foot. This forward static friction force is what helps propel walking or running.

Common MCAT forces include:

- Weight: gravitational force on a mass

$$
F_g = mg
$$

- Normal force: contact force perpendicular to a surface
- Tension: pulling force transmitted by a rope, tendon, or cable
- Kinetic friction: friction on sliding objects

$$
f_k = \mu_k N
$$

- Static friction: friction that prevents slipping, up to a maximum

$$
f_s \leq \mu_s N
$$

- Drag or viscous resistance: resistive forces from fluids, often opposing motion

> **Exam Warning:**
> Static friction is not automatically equal to $$\mu_sN$$. It adjusts as needed up to its maximum value. Use equality only when the object is just about to slip.

Free-body diagrams are the safest way to handle force questions. Draw the object as a point or box, include only forces acting on that object, resolve angled forces into components, and then apply Newton’s second law separately in each direction.

For an inclined plane, weight is often split into components parallel and perpendicular to the ramp:

$$
F_{\parallel} = mg\sin\theta
F_{\perp} = mg\cos\theta
$$

If the object does not accelerate perpendicular to the ramp, the normal force is usually:

$$
N = mg\cos\theta
$$

Friction then depends on this normal force, not necessarily on the full weight. This matters in passages about ramps, wheelchair movement, sliding objects, or body segments supported at an angle.

### Work, Energy, and Power in Biological Systems

Energy is the ability of a system to make something happen, including the ability to do work. The SI unit for energy is the joule. Work is not a separate form of energy; it is a process by which energy is transferred when a force acts through a displacement.

> **Key Term: work**
> Energy transferred by a force acting through a displacement.

For a constant force:

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

The angle is between the force and displacement. If the force is in the same direction as displacement, work is positive. If it opposes displacement, work is negative. If force is perpendicular to displacement, work is zero.

A classic example is carrying a heavy box horizontally at constant height. The upward force from the arms may be large, but if displacement is horizontal, the upward force does no mechanical work on the box. By contrast, lifting the box vertically requires positive work because the applied force and displacement are in the same direction.

> **Exam Warning:**
> Do not use $$W = Fd$$ unless the force is parallel to displacement. The cosine factor is often the difference between a correct answer and a trap answer.

Kinetic energy is energy of motion:

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

> **Key Term: kinetic energy**
> Energy associated with motion; it depends on mass and the square of speed.

Because speed is squared, doubling speed quadruples kinetic energy. This is a high-yield relationship for MCAT questions about collisions, moving fluids, falling objects, or sprinting.

Potential energy is stored energy associated with position or configuration. Gravitational potential energy near Earth’s surface is:

$$
U_g = mgh
$$

where height is measured relative to a chosen zero level, sometimes called a datum. Elastic potential energy in an ideal spring is:

$$
U_s = \frac{1}{2}kx^2
$$

where k is the spring constant and x is stretch or compression from equilibrium.

> **Key Term: potential energy**
> Stored energy associated with an object’s position, configuration, or interactions within a system.

Total mechanical energy is the sum of kinetic and potential energies:

$$
E_{\text{mech}} = K + U
$$

If only conservative forces act, total mechanical energy is conserved. Gravity and electrostatic force are conservative; ideal springs are treated as nearly conservative. Friction, air resistance, and viscous drag are nonconservative because they dissipate mechanical energy, commonly into thermal energy.

> **Key Term: conservative force**
> A force for which work is path independent and mechanical energy is not dissipated.
>
> **Key Term: nonconservative force**
> A force for which work depends on path and mechanical energy is dissipated into other forms, such as heat.

The work–energy theorem connects force and energy approaches:

$$
W_{\text{net}} = \Delta K
$$

More broadly, work can change kinetic energy, potential energy, thermal energy, or internal energy. In biological systems, chemical potential energy in ATP is converted into mechanical work by muscles, with some energy released as heat. This is why contracting muscles warm up during repeated use.

Power is the rate of energy transfer or work done:

$$
P = \frac{W}{t}
$$

> **Key Term: power**
> The rate at which work is done or energy is transferred; the SI unit is the watt.

A person who climbs stairs faster does the same approximate gravitational work as someone with the same mass climbing the same height, but has greater power output because the time is smaller.

### Equilibrium, Torque, and Mechanical Advantage

Equilibrium means no acceleration. For translational equilibrium:

$$
\sum F_x = 0
\sum F_y = 0
$$

For rotational equilibrium, net torque must also be zero:

$$
\sum \tau = 0
$$

> **Key Term: equilibrium**
> A state in which net force is zero and, when rotation is relevant, net torque is also zero.

Torque measures the tendency of a force to cause rotation:

$$
\tau = rF\sin\theta
$$

Here r is the distance from the pivot to where the force is applied, and theta is the angle between the lever arm and force. A longer lever arm or a more perpendicular force produces more torque.

> **Key Term: torque**
> The rotational effect of a force about an axis or pivot.

Human joints often act as levers. For example, when holding a weight in the hand with the forearm horizontal, the elbow is the pivot. The biceps may attach close to the elbow, giving it a short lever arm. Because torque equals force times lever arm, the biceps force can be much larger than the weight held in the hand. MCAT passages may give distances and ask for muscle force using torque balance.

> **Test Tip:**
> In torque problems, choose the pivot at a point where an unknown force acts. That unknown then has zero lever arm and produces no torque, simplifying the equation.

Static equilibrium requires both force and torque balance. A bridge, bone, or forearm can have zero net force but still rotate if torques are unbalanced. Conversely, torques can balance while net force is nonzero, causing translational acceleration.

Mechanical advantage describes how simple machines reduce the input force needed to perform work. The tradeoff is distance: if a machine reduces required force, the input force usually must be applied over a greater distance.

> **Key Term: mechanical advantage**
> The factor by which a machine multiplies input force to help accomplish work.

For an ideal machine with no energy loss:

$$
W_{\text{in}} = W_{\text{out}}
F_{\text{in}}d_{\text{in}} = F_{\text{out}}d_{\text{out}}
$$

Real machines lose some mechanical energy to friction, deformation, or heat, so work input exceeds useful work output.

Inclined planes reduce force by increasing distance. A ramp used to raise a wheelchair to a height h requires less force than lifting vertically, but the person pushes over a longer distance. Pulleys can reduce effort force by increasing the length of rope pulled. Levers can multiply force when the effort arm is longer than the resistance arm.

The six classical simple machines are the inclined plane, wedge, wheel and axle, lever, pulley, and screw. The MCAT is most likely to test ramps, pulleys, and levers because they connect naturally to body mechanics, patient movement, and laboratory equipment.

A quick ramp example: raising a 50 kg object by 2 m increases gravitational potential energy by approximately:

$$
\Delta U_g = mgh = (50)(10)(2) = 1000\ \mathrm{J}
$$

If a frictionless ramp is 10 m long, the required average input force along the ramp is:

$$
F = \frac{W}{d} = \frac{1000}{10} = 100\ \mathrm{N}
$$

Lifting vertically would require about 500 N, but over only 2 m. The ideal work is the same; the force–distance tradeoff changes.

## Key Point Checklist

This article has covered the following key knowledge points:

- Displacement and velocity are vectors; distance and speed are scalars.
- Acceleration is any change in velocity, including a change in direction.
- Newton’s second law uses net force: $$\sum F = ma$$.
- Weight is $$mg$$, while normal force depends on the surface and situation.
- Kinetic friction equals $$\mu_kN$$, but static friction is variable up to $$\mu_sN$$.
- Work is energy transfer by force through displacement: $$W = Fd\cos\theta$$.
- Kinetic energy depends on speed squared: $$K = \frac{1}{2}mv^2$$.
- Gravitational potential energy near Earth is $$U_g = mgh$$.
- If only conservative forces act, total mechanical energy is conserved.
- Nonconservative forces such as friction and drag dissipate mechanical energy.
- Equilibrium requires zero net force; static rigid-body equilibrium also requires zero net torque.
- Simple machines reduce required force by increasing input distance and may lose energy through inefficiency.

## Key Terms and Concepts

- translational motion
- displacement
- velocity
- force
- work
- kinetic energy
- potential energy
- conservative force
- nonconservative force
- power
- equilibrium
- torque
- mechanical advantage


Foundational concept 4: Physical principles - 4A: Translational motion, forces, work, energy, and equilibrium