## MCAT Physics and Math Review

**Chapter 1: Kinematics and Dynamics**

### Conclusion

In this chapter, we’ve equipped you with the math—the language of physics—necessary to understand our first important topic for the MCAT *Chemical and Physical Foundations of Biological Systems* section: kinematics and Newtonian mechanics. This study of objects in motion allows us to describe an object’s position, displacement, distance traveled, velocity, speed, and acceleration with respect to time. We now understand how to use the four key kinematics equations when objects experience constant acceleration, a relatively simple scenario presented on Test Day.

We also learned that different kinds of forces act on objects to cause them to move in certain ways. Application of forces may cause objects to accelerate or decelerate according to Newton’s second law. If the vector sum of all the forces acting on an object is equal to zero, the forces cancel out, and the object experiences no acceleration, a condition known as translational equilibrium. This is expressed in Newton’s first law. Even when objects aren’t touching, they can still exert forces between them, as described by Newton’s third law. We considered linear motion, projectile motion, inclined planes, and circular motion. We also considered the special conditions of translational and rotational equilibrium.

We hope that you will come to appreciate the relevance that these concepts and principles have for your performance not only on the MCAT but also in medical school, residency training, and your career as a physician. Your careful consideration of the discussion topics in this chapter and your practice with the kinds of problems demonstrated here earn you many points on Test Day.

### Concept Summary

*Units*

· The MCAT will test the **SI units** that are related to the metric system. The SI units include meter, kilogram, second, ampère, mole, kelvin, and candela.

*Vectors and Scalars*

· **Vectors** are physical quantities that have both magnitude and direction. Vector quantities include displacement, velocity, acceleration, and force, among others.

· **Scalars** are quantities without direction. Scalar quantities may be the magnitude of vectors, like speed, or may be dimensionless, like coefficients of friction.

· Vector addition may be accomplished using the tip-to-tail method or by breaking a vector into its components and using the Pythagorean theorem.

· Vector subtraction is accomplished by changing the direction of the subtracted vector and then following the procedures for vector addition.

· Multiplying a vector by a scalar changes the magnitude and may reverse the direction.

· Multiplying two vectors using the **dot product** results in a scalar quantity. The dot product is the product of the vectors’ magnitudes and the cosine of the angle between them.

· Multiplying two vectors using the **cross product** results in a vector quantity. The cross product is the product of the vectors’ magnitudes and the sine of the angle between them. The right-hand rule is used to determine the resultant vector’s direction.

*Displacement and Velocity*

· **Displacement** is the vector representation of a change in position. It is path independent and is equivalent to the straight line distance between the start and end locations.

· **Distance** is a scalar quantity that reflects the path traveled.

· **Velocity** is the vector representation of the change in displacement with respect to time.

o **Average velocity** is the total displacement divided by the total time.

o **Average speed** is the total distance traveled divided by the total time.

o **Instantaneous velocity** is limit of the change in displacement over time as the change in time approaches zero.

o **Instantaneous speed** is the magnitude of the instantaneous velocity vector.

*Forces and Acceleration*

· A **force **is any push or pull that has the potential to result in an acceleration.

· **Gravity** is the attractive force between two objects as a result of their masses.

· **Friction** is a force that opposes motion as a function of electrostatic interactions at the surface between two objects.

o **Static friction** exists between two objects that are not in motion relative to each other.

o **Kinetic friction** exists between two objects that are in motion relative to each other.

o Whereas static friction can take on many values depending on the magnitude of an applied force, kinetic friction is a constant value.

o The **coefficient of friction** depends on the two materials in contact. The coefficient of static friction is always higher than the coefficient of kinetic friction.

· Mass and weight are not synonymous.

o **Mass** is a measure of the inertia of an object—its amount of material.

o **Weight **is the force experienced by a given mass due to the gravitational attraction to the Earth.

· **Acceleration** is the vector representation of the change in velocity over time. Average or instantaneous acceleration may both be considered, similar to velocity.

*Newton’s Laws*

· **Newton’s first law**, or the **law of inertia**, states that an object will remain at rest or move with a constant velocity if there is no net force on the object.

· **Newton’s second law** states that any acceleration is the result of the sum of the forces acting on the object and its mass.

· **Newton’s third law** states that any two objects interacting with one another experience equal and opposite forces as a result of their interaction.

*Motion with Constant Acceleration*

· **Linear motion** includes **free fall** and motion in which the velocity and acceleration vectors are parallel or antiparallel.

· **Projectile motion** contains both an *x*- and *y-*component. Assuming negligible air resistance, the only force acting on the object is gravity.

· **Inclined planes** are another example of two-dimensional movement. It is often easiest to consider the dimensions as being parallel and perpendicular to the surface of the plane.

· **Circular motion** is best thought of as having radial and tangential dimensions. In **uniform circular motion**, the only force is the **centripetal force**, pointing radially inward. The instantaneous velocity vector always points tangentially.

*Mechanical Equilibrium*

· **Free body diagrams** are representations of the forces acting on an object. They are useful for equilibrium and dynamics problems.

· **Translational equilibrium** occurs in the absence of any net forces acting on an object. An object in translational equilibrium has a constant velocity, and may or may not also be in rotational equilibrium.

· **Rotational equilibrium** occurs in the absence of any net **torques** acting on an object. Rotational motion may consider any pivot point, but the center of mass is most common. An object in rotational equilibrium has a constant angular velocity; on the MCAT, the angular velocity is usually zero.

### Answers to Concept Checks

· **1.1**

1. Force will obey the same relationship with mass and acceleration, regardless of the unit system. Force is always the product of mass and acceleration, so one pound (lb) must be equal to one

2. ångström < centimeter < inch < foot < mile

· **1.2**

1. Vector addition, unlike vector multiplication, is a commutative function. The resultant of **A **+ **B** is the same as **B **+ **A**, so there would be no difference between the two resultants.

2. Vector subtraction, like vector multiplication, is not a commutative function. The resultant of **A **– **B** has the same magnitude as **B **– **A**, but is oriented in the opposite direction.

3. A scalar is calculated from two vectors by using the dot product: **A** **·** **B** = |**A**| |**B**| cos*θ*. A vector is calculated by using the cross product: **A** × **B** = |**A**| |**B**| sin*θ*.

4. False. This would be true of an addition problem in which both vectors have equal magnitude, but it is never true for vector multiplication. To find the direction of **C**, we must use the right-hand rule. If the thumb points in the direction of **A**, and the fingers point in the direction of **B**, then our palm, **C**, points out of the page.

· **1.3**

1. Instantaneous speed is the magnitude of the instantaneous velocity vector. Average speed and average velocity may be unrelated because speed does not depend on displacement, but is rather the total distance traveled divided by time.

2. True. Displacement considers the most direct route between two points. Distance will always be equal or larger in magnitude than displacement.

3. Velocity is the rate of the change of the displacement of an object. Displacement is a function of velocity acting over a period of time.

· **1.4**

1. The direction of the frictional force always opposes movement. Once the instantaneous velocity vector is known (or net force, in the case of static friction), the frictional force must be in the opposite direction.

2. If there is no net force acting on an object, then that object is not experiencing an acceleration and it has a constant velocity.

3. False. Forces are always reciprocal in nature. When the Earth creates a force on a person, the person also exerts a force of the same magnitude on the Earth (in the opposite direction). The difference in masses gives the Earth an apparent acceleration of zero.

4. Gravity and frictional forces were discussed in this chapter. Electrostatic, magnetic, elastic, weak nuclear, and strong nuclear forces are other examples of forces.

· **1.5**

1. Any answer which is similar to the following is acceptable:

1. In the absence of any forces—or when the total force is zero—there will be no change in velocity.

2. Acceleration results from the sum of the force vectors.

3. For any two interacting objects, all forces acting on one object have an equal and opposing force acting on the other object.

· **1.6**

1. The only force acting in both free fall and projectile motion is gravity.

2. The product of sine and cosine is maximized when the angle is 45°. Because horizontal displacement relies on both measurements, the maximum horizontal displacement will also be achieved at this angle. Vertical displacement will always be zero as the object returns to the starting point. Objects launched vertically will experience the greatest vertical *distance*.

3. If the equation for centripetal force is and force is simply mass times acceleration (from Newton’s second law), then

· **1.7**

1. A moving object can be in either translational or rotational equilibrium (or both). Translational equilibrium only requires the net force on an object be zero—its velocity is constant. The corresponding condition in rotational equilibrium is that net torque equals zero—its angular velocity is constant.

2. One could place the fulcrum one quarter of the way across the lever, closer to the object. The ratio of the lever arms would then be 3:1, which means that only one-third of the original force is necessary. (Alternatively, the fulcrum could be placed at the end with the object one-third of the way across the lever. This would again result in a 3:1 ratio of lever arms, meaning that only one-third of the original force is necessary.)

### Equations to Remember

(1.1) **Component vectors**:

(1.2) **Pythagorean theorem**: **X**^{2} + **Y**^{2} = **V**^{2} or

(1.3) **Determination of direction from component vectors**:

(1.4) **Dot product**: **A · B** = |**A**| |**B**| cos*θ*

(1.5) **Cross product**: **A** × **B** = |**A**| |**B**| sin*θ*

(1.6) **Instantaneous velocity**:

(1.7) **Average velocity**:

(1.8) **Universal gravitation equation**:

(1.9) **Static friction**: 0 ≤ *f*_{s} ≤ *μ*_{s}*N*

(1.10) **Kinetic friction**: *f*_{k} = *μ*_{k}*N*

(1.11) **Force of gravity (weight on Earth)**: **F**_{g} = *m***g**

(1.12) **Center of mass**:

(1.13) **Average acceleration**:

(1.14) **Instantaneous acceleration**:

(1.15) **Newton’s first law**: **F**_{net} = *m***a** = 0

(1.16) **Newton’s second law**: **F**_{net} = *m***a**

(1.17) **Newton’s third law**: **F**_{AB} = –**F**_{BA}

(1.18) **Kinematics (no displacement)**: **v = v**_{0} + **a***t*

(1.19) **Kinematics (no final velocity)**:

(1.20) **Kinematics (no time)**: **v**^{2} = **v**_{0}^{2} + 2**ax**

(1.21) **Kinematics (no acceleration)**:

(1.22) **Components of gravity on an inclined plane**:

(1.23) **Centripetal force**:

(1.24) **Torque**: ** τ** =

**r**×

**F**=

*rF*sin

*θ*

### Shared Concepts

· **General Chemistry Chapter 1**

o Atomic Structure

· **General Chemistry Chapter 3**

o Bonding and Chemical Interactions

· **Physics and Math Chapter 2**

o Thermodynamics

· **Physics and Math Chapter 4**

o Fluids

· **Physics and Math Chapter 5**

o Electrostatics and Magnetism

· **Physics and Math Chapter 10**

o Mathematics