功、能与功率

• 3.1.A.1 An object's translational kinetic energy is given by the equation
  • Equation: $K = \dfrac{1}{2}mv^2$
• 3.1.A.2 Translational kinetic energy is a scalar quantity.
• 3.1.A.3 Different observers may measure different values of the translational kinetic energy of an object, depending on the observer's frame of reference.

• 3.2.A.1 Work is the amount of energy transferred into or out of a system by a force exerted on that system over a distance.
  • 3.2.A.1.i The work done by a conservative force exerted on a system is path-independent and only depends on the initial and final configurations of that system.
  • 3.2.A.1.ii The work done by a conservative force on a system—or the change in the potential energy of the system—will be zero if the system returns to its initial configuration.
  • 3.2.A.1.iii Potential energies are associated only with conservative forces.
  • 3.2.A.1.iv The work done by a nonconservative force is path-dependent.
  • 3.2.A.1.v The most common nonconservative forces are friction and air resistance.
• 3.2.A.2 Work is a scalar quantity that may be positive, negative, or zero.
• 3.2.A.3 The work done on an object by a variable force is calculated using
  • Equation: $W = \displaystyle\int_a^b \vec{F}(r) \cdot d\vec{r}$, where the integral is taken over the path from point $a$ to point $b$.
  • 3.2.A.3.i The dot product between two vectors, $\vec{A}$ and $\vec{B}$, results in a scalar quantity of magnitude $\vec{A} \cdot \vec{B} = AB\cos\theta$.
  • 3.2.A.3.ii Only the component of the force exerted on a system that is parallel to the displacement of the point of application of the force will change the system's total energy.
  • 3.2.A.3.iii If the component of the force exerted on a system that is parallel to the displacement is constant, the work done on the system by the force is given by the derived equation $W = F_{\parallel}d = Fd\cos\theta$.
  • 3.2.A.3.iv The component of the force exerted on a system perpendicular to the direction of the displacement of the system's center of mass can change the direction of the system's motion without changing the system's kinetic energy.
• 3.2.A.4 The work–energy theorem states that the change in an object's kinetic energy is equal to the sum of the work (net work) being done by all forces exerted on the object.
  • Equation: $\Delta K = \displaystyle\sum W_i = \sum F_{\parallel,i}\, d_i$
  • 3.2.A.4.i An external force may change the configuration of a system. The component of the external force parallel to the displacement times the displacement of the point of application of the force gives the change in kinetic energy of the system.
  • 3.2.A.4.ii If the system's center of mass and the point of application of the force move the same distance when a force is exerted on a system, then the system may be modeled as an object, and only the system's kinetic energy can change.
  • 3.2.A.4.iii The energy dissipated by friction is typically equated to the force of friction times the length of the path over which the force is exerted.
    • Equation: $\Delta E_{\text{mech}} = F_f d\cos\theta$
  • 3.2.A.5 Work is equal to the area under the curve of a graph of $F_{\parallel}$ as a function of displacement.

Boundary statement: AP Physics C: Mechanics only expects students to analyze the transfer of mechanical energy, although students should be aware that mechanical energy may be dissipated in the form of thermal energy or sound.

• 3.3.A.1 A system composed of two or more objects has potential energy if the objects within that system only interact with each other through conservative forces.
• 3.3.A.2 Potential energy is a scalar quantity associated with the position of objects within a system.
• 3.3.A.3 The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.
• 3.3.A.4 The relationship between conservative forces exerted on a system and the system's potential energy is
  • Equation: $\Delta U = -\displaystyle\int_a^b \vec{F}_{cf}(r) \cdot d\vec{r}$
• 3.3.A.5 The conservative forces exerted on a system in a single dimension can be determined using the slope of the system's potential energy with respect to position in that dimension; these forces point in the direction of decreasing potential energy.
  • Equation: $F_x = -\dfrac{dU(x)}{dx}$
• 3.3.A.6 Graphs of a system's potential energy as a function of its position can be useful in determining physical properties of that system.
  • 3.3.A.6.i Stable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object opposite to the direction of the small displacement, accelerating the object back toward the equilibrium position.
  • 3.3.A.6.ii Unstable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object in the same direction as the small displacement, accelerating the object away from the equilibrium position.
  • 3.3.A.6.iii In a given dimension, stable equilibrium positions exist at locations where the potential energy as a function of position in that dimension has a local minimum.
  • 3.3.A.6.iv In a given dimension, unstable equilibrium positions occur at locations where the potential energy as a function of position in that dimension has a local maximum.
• 3.3.A.7 The potential energy of common physical systems can be described using the physical properties of that system.
  • 3.3.A.7.i The elastic potential energy of an ideal spring is given by the following equation, where $\Delta x$ is the distance the spring has been stretched or compressed from its equilibrium length.
    • Equation: $U_s = \dfrac{1}{2}k(\Delta x)^2$
  • 3.3.A.7.ii The general form for the gravitational potential energy of a system consisting of two approximately spherical distributions of mass (e.g., moons, planets, or stars) is given by the equation
    • Equation: $U_g = -G\dfrac{m_1 m_2}{r}$
  • 3.3.A.7.iii Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy in a system consisting of an object with mass $m$ and a planet with gravitational field of magnitude $g$ when the object is near the surface of the planet may be approximated by the equation
    • Equation: $\Delta U_g = mg\Delta y$
• 3.3.A.8 The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system.

• 3.4.A.1 A system composed of only a single object can only have kinetic energy.
• 3.4.A.2 A system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies.

• 3.4.B.1 Mechanical energy is the sum of a system's kinetic and potential energies.
• 3.4.B.2 Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.
• 3.4.B.3 A system may be selected so that the total energy of that system is constant.
• 3.4.B.4 If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.

• 3.4.C.1 Energy is conserved in all interactions.
• 3.4.C.2 If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant.
• 3.4.C.3 If the work done on a selected system is nonzero, energy is transferred between the system and the environment.

Boundary statement: AP Physics C: Mechanics expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

• 3.5.A.1 Power is the rate at which energy changes with respect to time, either by transfer into or out of a system or by conversion from one type to another within a system.
• 3.5.A.2 Average power is the amount of energy being transferred or converted, divided by the time it took for that transfer or conversion to occur.
  • Equation: $P_{\text{avg}} = \dfrac{\Delta E}{\Delta t}$
• 3.5.A.3 Because work is the change in energy of an object or system due to a force, average power is the total work done, divided by the time during which that work was done.
  • Equation: $P_{\text{avg}} = \dfrac{W}{\Delta t}$
• 3.5.A.4 The instantaneous power delivered to an object by a force is given by the equation
  • Equation: $P_{\text{inst}} = \dfrac{dW}{dt}$
• 3.5.A.5 The instantaneous power delivered to an object by the component of a constant force parallel to the object's velocity can be described with the derived equation
  • Equation: $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$