AP Physics 1 Equation Sheet for 2024 Exam

$\vec{F}_{net}=\vec{F}_1+\vec{F}_2+\cdots$ is the vector sum of all forces acting on the object.

 Newton's second law for uniform ($v=constant$) circular motion. 

For such motion, a force is needed to keep an object moving around a circle.

The direction of this force is always toward the center of the circle.

Examples of these forces are the force of gravity (as in the motion of a satellite or moon around the Earth), the tension in a cord, or a component of normal force (as the motion of a car rounding a banked curve).

The magnitude of the attractive gravitational force that an object of $m_1$ applies to an object of mass $m_2$ positioned by a center-to-center distance $r$.

$G=6.67\times 10^{-11}\,{\rm N\cdot m^2/kg^2}$ is a universal constant.

This formula is used primarily to find the magnitude of the force that the Sun exerts on planets, between two planets, or between a planet and its moon. In the AP Physics 1 exam, usually, the order of magnitude estimates for the force will be asked.

The amount of gravitational field $g$ produced by a planet of mass $M$ at distance $d$ from the planet's center. 

Substituting $M_{planet}=M_{earth}$ and $d=r_{earth}$ gives the value of free-fall acceleration at the surface of the Earth, $g=9.8\,\rm m/s^2$.

The work done by a constant force $\vec{F}$ on an object while its displacement is $\Delta \vec{x}$. 

$F$ is the magnitude of the applied force in newtons, $d$ is the magnitude of the displacement in meters, and $\theta$ is the angle between the direction of $\vec{F}$ and the direction of $\Delta \vec{x}$.

The sign of $\cos\theta$ determines the sign of the work. 

The SI unit of work is the joule: $\rm 1\,J=1\,N\cdot m$.

Work is a scalar quantity in physics. 

Positive work: the applied force and the object's displacement are in the same direction, $\theta=0\rightarrow \cos\theta=1$. The system gains energy. 

Negative work: the applied force and the object's displacement are in the opposite direction, $\theta=180^\circ \rightarrow \cos\theta=-1$. The system loses energy. Friction force always does negative work.

Zero work: the external force points perpendicular to the object's displacement, $\theta=90^\circ \rightarrow \cos\theta=0$. There is no change in the energy of the system, like the work done by normal force.

The gravitational potential energy of an object of mass $m$ at height $h$ (in meters) near the Earth's surface. 

The gravitational potential energy equation for to two objects located far away in space, with infinity taken as zero potential.

A single object cannot have gravitational potential energy.

(Translational) kinetic energy is the energy of an object of mass moving at a speed of $v$. It is a scalar quantity. SI unit: joules ($\rm J$).

Notice that kinetic energy is always a positive value.

Elastic potential energy is the energy of a stretched or compressed spring as much $x$ from its equilibrium position.

$k$ is the spring stiffness constant with an SI unit of $\rm N/m$.

The work-energy theorem states that the net work done on an object of mass $m$ equals the change in its kinetic energy or that the work done by a non-conservative force $W_{nc}$ changes the total mechanical energy of a system ($E_{total}=U+K$). 

The friction force is an example of a non-conservative force, meaning the work done by it depends on the path taken.

Power is defined as the rate of energy transfer, or the rate of work done on a system during a time interval of $\Delta t$. 

SI unit: the watt, ($W=\rm J/s$)

 Impulse is the product of average external constant force $\vec{F}$ exerted on an object during a time interval $\Delta t$ and that time interval. 

SI unit: $\rm N\cdot s$.

 The impulse-momentum theorem states that the impulse of a constant force delivered to an object equals the change in momentum of the object.

The area under the force vs. time graph is the impulse. 

When no net external forces act on an isolated system, the total momentum of that system before an interaction is the same as after the interaction. This principle is called the conservation of momentum. 

In all collisions, momentum is a conserved quantity.

In an inelastic collision, the momentum is conserved, but kinetic energy is not. 

$\Delta U_{int}$ is the energy of motion and interaction of microscopic particles making up the object, known as internal energy or heat energy. 

In a totally inelastic collision, the colliding objects stick together and have a common speed after collision.

The period, $T$, is the time required to complete one back-and-forth cycle. 

The frequency, $f$, is the number of cycles per second. These are defined for a general periodic motion. 

SI unit of frequency is the hertz ($\rm Hz$), where $\rm 1\,Hz= 1\,s^{-1}$.

The period of a simple pendulum of length $\ell$. 

Note that the period of a simple pendulum depends only on its length and on the magnitude of the gravitational constant near the surface of that planet, like the Earth $g=9.8\,\rm m/s^2$. 

This period does not depend on the mass of the object hanging at the end of the pendulum.

The average rotational (angular) velocity of a revolving body. It is measured in $\rm rad/s$ or revolutions per minute ($\rm rpm$).

Angular velocity is a vector quantity in physics, meaning it has both a magnitude and a direction. 

Rotational (angular) speed is the magnitude of rotational velocity. 

The sign of $\omega$ is conventionally taken to be positive for counterclockwise turning, and negative for clockwise turning.

One revolution ($\rm rev$) corresponds to a complete rotation around a circle. \[\rm 1\,rev=360^\circ \]

Kinematic equations for rotational motion with constant angular acceleration.

For simplicity in the AP Physics 1 questions, choose $\theta_0=0$. 

$\omega _0$ is the initial angular velocity at the initial time $t_0=0$. 

$\omega$ and $\theta$ are the angular position and velocity, respectively, at time $t$.

$\bar{\omega}$ is the average angular velocity.

The torque provided by a force $\vec{F}$ exerted on an object at distance $r$ from its rotation axis. 

$r_{\bot}=r\sin\theta$ is called the lever arm and is defined as the perpendicular distance from the axis of rotation to the line along which the force acts. 

$F_{\bot}$ is the component of the force perpendicular to $r$, the distance between the point of application of the force and the pivot point.

$\theta$ is the angle between $\vec{F}$ and $r$.

SI unit: Newton-meter ($\rm N\cdot m$)

The product of rotational inertia $I$ on an object and its angular velocity $\omega$ is defined as angular momentum. 

Angular momentum is a vector quantity in physics. 

In the topic of rotational motion, the direction of vector quantities is usually shown with positive and negative signs.

$L$ is positive for counterclockwise rotation and negative for clockwise rotation. 

It is analogous to linear momentum, $\vec{p}=m\vec{v}$.

It is necessary to define a rotational axis to find the angular momentum.

When the net torque acting on a rotating object is zero, the total angular momentum of it remains constant. This is known as the conservation of angular momentum.

For an extended object, $L=I\omega$. 

For a point particle of mass $m$ moving at a speed of $v$, we have $L=mvr$, where $r$ is the closest distance of approach.

The angular acceleration $\alpha$ of an object equals the net torque produced by forces exerted on the object divided by its rotational inertia, $I$.

This is also known as Newton's second law for rotation.