Force and Translational Dynamics · ⁨力と並進運動の力学⁩

Learning Objective 2.1.A: Describe the properties and interactions of a system.

• 2.1.A.1 System properties are determined by the interactions between objects within the system.
• 2.1.A.2 If the properties or interactions of the constituent objects within a system are not important in modeling the behavior of the macroscopic system, the system can itself be treated as a single object.
• 2.1.A.3 Systems may allow interactions between constituent parts of the system and the environment, which may result in the transfer of energy or mass.
• 2.1.A.4 Individual objects within a chosen system may behave differently from each other as well as from the system as a whole.
• 2.1.A.5 The internal structure of a system affects the analysis of that system.
• 2.1.A.6 As variables external to a system are changed, the system's substructure may change.

Learning Objective 2.1.B: Describe the location of a system's center of mass with respect to the system's constituent parts.

• 2.1.B.1 For objects or systems with symmetrical mass distributions, the center of mass is located on lines of symmetry.
• 2.1.B.2 The location of a system's center of mass along a given axis can be calculated using the equation
  • Equation: $\vec{x}_{\text{cm}} = \dfrac{\sum m_i \vec{x}_i}{\sum m_i}$
• 2.1.B.3 For a nonuniform solid that can be considered as a collection of differential masses, $dm$, the solid's center of mass can be calculated using the equation
  • Equation: $\vec{r}_{\text{cm}} = \dfrac{\int \vec{r}\, dm}{\int dm}$
  • 2.1.B.3.i The linear mass density of a rod or other linear rigid body is the derivative of the rod's mass with respect to the position of the differential mass element on the rigid body.
    • Equation: $\lambda = \dfrac{d}{d\ell} m(\ell)$
  • 2.1.B.3.ii If a function of mass density is given for a solid, the total mass can be determined by integrating the mass density over the length (one dimension), area (two dimensions), or volume (three dimensions) of the solid. For example:
    • Equation: $M_{\text{total}} = \int \rho(r)\, dV$
• 2.1.B.4 A system can be modeled as a singular object that is located at the system's center of mass.

Learning Objective 2.2.A: Describe a force as an interaction between two objects or systems

• 2.2.A.1 Forces are vector quantities that describe the interactions between objects or systems.
  • 2.2.A.1.i A force exerted on an object or system is always due to the interaction of that object or system with another object or system.
  • 2.2.A.1.ii An object or system cannot exert a net force on itself.
• 2.2.A.2 Contact forces describe the interaction of an object or system touching another object or system and are macroscopic effects of interatomic electric forces.

Learning Objective 2.2.B: Describe the forces exerted on an object or system using a free-body diagram.

• 2.2.B.1 Free-body diagrams are useful tools for visualizing forces being exerted on a single object or system and for determining the equations that represent a physical situation.
• 2.2.B.2 The free-body diagram of an object or system shows each of the forces exerted on the object or system by the environment.
• 2.2.B.3 Forces exerted on an object or system are represented as vectors originating from the representation of the center of mass, such as a dot. A system is treated as though all of its mass is located at the center of mass.
• 2.2.B.4 A coordinate system with one axis parallel to the direction of acceleration of the object or system simplifies the translation from free-body diagram to algebraic representation. For example, in a free-body diagram of an object on an inclined plane, it is useful to set one axis parallel to the surface of the incline.

Boundary statement: AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect students to depict the forces exerted on objects, not the force components on free-body diagrams. On the AP Physics exams, individual forces represented on a free-body diagram must be drawn as individual straight arrows, originating on the dot and pointing in the direction of the force. Individual forces that are in the same direction must be drawn side by side, not overlapping.

Learning Objective 2.3.A: Describe the interaction of two objects or systems using Newton's third law and a representation of paired forces exerted on each object or system.

• 2.3.A.1 Newton's third law describes the interaction of two objects or systems in terms of the paired forces that each exerts on the other.
  • Equation: $\vec{F}_{\text{A on B}} = -\vec{F}_{\text{B on A}}$
• 2.3.A.2 Interactions between objects within a system (internal forces) do not influence the motion of a system's center of mass.
• 2.3.A.3 Tension is the macroscopic net result of forces that infinitesimal segments of a string, cable, chain, or similar system exert on each other in response to an external force.
  • 2.3.A.3.i An ideal string has negligible mass and does not stretch when under tension.
  • 2.3.A.3.ii The tension in an ideal string is the same at all points within the string.
  • 2.3.A.3.iii In a string with nonnegligible mass, tension may not be the same at all points within the string.
  • 2.3.A.3.iv An ideal pulley is a pulley that has negligible mass and rotates about an axle through its center of mass with negligible friction.

Learning Objective 2.4.A: Describe the conditions under which a system's velocity remains constant.

• 2.4.A.1 The net force on a system is the vector sum of all forces exerted on the system.
• 2.4.A.2 Translational equilibrium is the configuration of forces such that the net force exerted on a system is zero.
  • Derived equation: $\sum \vec{F}_i = 0$
• 2.4.A.3 Newton's first law states that if the net force exerted on a system is zero, the velocity of that system will remain constant.
• 2.4.A.4 Forces may be balanced in one dimension but unbalanced in another. The system's velocity will change only in the direction of the unbalanced force.
• 2.4.A.5 An inertial reference frame is one from which an observer would verify Newton's first law of motion.

Learning Objective 2.5.A: Describe the conditions under which a system's velocity changes.

• 2.5.A.1 Unbalanced forces are a configuration of forces such that the net force exerted on a system is not equal to zero.
• 2.5.A.2 Newton's second law of motion states that the acceleration of a system's center of mass has a magnitude proportional to the magnitude of the net force exerted on the system and is in the same direction as that net force.
  • Equation: $\vec{a}_{\text{sys}} = \dfrac{\sum \vec{F}}{m_{\text{sys}}} = \dfrac{\vec{F}_{\text{net}}}{m_{\text{sys}}}$
• 2.5.A.3 The velocity of a system's center of mass will only change if a nonzero net external force is exerted on that system.

Learning Objective 2.6.A: Describe the gravitational interaction between two objects or systems with mass.

• 2.6.A.1 Newton's law of universal gravitation describes the gravitational force between two objects or systems as directly proportional to each of their masses and inversely proportional to the square of the distance between the systems' centers of mass.
  • Equation: $\left| \vec{F}_g \right| = G \dfrac{m_1 m_2}{r^2}$
  • 2.6.A.1.i The gravitational force is attractive.
  • 2.6.A.1.ii The gravitational force is always exerted along the line connecting the center of mass of the two interacting systems.
  • 2.6.A.1.iii The gravitational force on a system can be considered to be exerted on the system's center of mass.
• 2.6.A.2 A field models the effects of a noncontact force exerted on an object at various positions in space.
  • 2.6.A.2.i The magnitude of the gravitational field created by a system of mass $M$ at a point in space is equal to the ratio of the gravitational force exerted by the system on a test object of mass $m$ to the mass of the test object.
    • Derived equation: $\left| \vec{g} \right| = \dfrac{\left| \vec{F}_g \right|}{m} = G \dfrac{M}{r^2}$
  • 2.6.A.2.ii If the gravitational force is the only force exerted on an object, the observed acceleration of the object (in $\text{m/s}^2$) is numerically equal to the magnitude of the gravitational field strength (in $\text{N/kg}$) at that location.
• 2.6.A.3 The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight.
  • Derived equation: $\text{Weight} = F_g = mg$

Learning Objective 2.6.B: Describe situations in which the gravitational force can be considered constant.

• 2.6.B.1 If the gravitational force between two systems' centers of mass has a negligible change as the relative position of the two systems changes, the gravitational force can be considered constant at all points between the initial and final positions of the systems.
• 2.6.B.2 Near the surface of Earth, the strength of the gravitational field is
  • Equation: $g \approx 10\ \text{N/kg}$

Learning Objective 2.6.C: Describe the conditions under which the magnitude of a system's apparent weight is different from the magnitude of the gravitational force exerted on that system.

• 2.6.C.1 The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system.
• 2.6.C.2 If the system is accelerating, the apparent weight of the system is not equal to the magnitude of the gravitational force exerted on the system.
• 2.6.C.3 A system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system.
• 2.6.C.4 The equivalence principle states that an observer in a noninertial reference frame is unable to distinguish between an object's apparent weight and the gravitational force exerted on the object by a gravitational field.

Learning Objective 2.6.D: Describe inertial and gravitational mass.

• 2.6.D.1 Objects have inertial mass, or inertia, a property that determines how much an object's motion resists changes when interacting with another object.
• 2.6.D.2 Gravitational mass is related to the force of attraction between two systems with mass.
• 2.6.D.3 Inertial mass and gravitational mass have been experimentally verified to be equivalent.

Learning Objective 2.6.E: Describe the gravitational force exerted on an object by a uniform spherical distribution of mass.

• 2.6.E.1 The net gravitational force exerted on an object by a uniform spherical distribution of mass is the sum of the individual forces from small differential masses that comprise the distribution.
• 2.6.E.2 Newton's shell theorem describes the net gravitational force exerted on an object by a uniform spherical shell of mass.
  • 2.6.E.2.i The net gravitational force exerted on an object inside a thin spherical shell is zero.
  • 2.6.E.2.ii The net gravitational force exerted on an object outside a thin spherical shell can be determined by treating the shell as a single massive object located at the center of the shell.
  • 2.6.E.2.iii An object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere.
  • 2.6.E.2.iv The partial mass of a sphere that contributes to the net gravitational force exerted on an object within that sphere is the portion of the sphere's mass located a distance less than or equal to the object's distance from the center of the sphere and can be calculated using the density of the sphere.
    • Derived equation: $m_{\text{partial}} = \rho \dfrac{4}{3} \pi \left( r_{\text{partial}} \right)^3$
• 2.6.E.3 The gravitational force exerted on an object within a uniform sphere can be shown to be proportional to the object's distance from the sphere's center.
  • Derived equation: $F_{g,\text{partial}} = -k r_{\text{partial}}$

Boundary statement: AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton's shell theorem.

Learning Objective 2.7.A: Describe kinetic friction between two surfaces.

• 2.7.A.1 Kinetic friction occurs when two surfaces in contact move relative to each other.
  • 2.7.A.1.i The kinetic friction force is exerted in a direction opposite the motion of each surface relative to the other surface.
  • 2.7.A.1.ii The force of friction between two surfaces does not depend on the size of the surface area of contact.
• 2.7.A.2 The magnitude of the kinetic friction force exerted on an object is the product of the normal force the surface exerts on the object and the coefficient of kinetic friction.
  • Equation: $\left| \vec{F}_{f,k} \right| = \left| \mu_k \vec{F}_N \right|$
  • 2.7.A.2.i The coefficient of kinetic friction depends on the material properties of the surfaces that are in contact.
  • 2.7.A.2.ii Normal force is the perpendicular component of the force exerted on an object by the surface with which it is in contact; it is directed away from the surface.

Learning Objective 2.7.B: Describe static friction between two surfaces.

• 2.7.B.1 Static friction may occur between the contacting surfaces of two objects that are not moving relative to each other.
• 2.7.B.2 Static friction adopts the value and direction required to prevent an object from slipping or sliding on a surface.
  • Equation: $\left| \vec{F}_{f,s} \right| \leq \left| \mu_s \vec{F}_n \right|$
  • 2.7.B.2.i Slipping and sliding refer to situations in which two surfaces are moving relative to each other.
  • 2.7.B.2.ii There exists a maximum value for which static friction will prevent an object from slipping on a given surface.
    • Derived equation: $F_{f,s,\text{max}} = \mu_s F_N$
• 2.7.B.3 The coefficient of static friction is typically greater than the coefficient of kinetic friction for a given pair of surfaces.

Learning Objective 2.8.A: Describe the force exerted on an object by an ideal spring.

• 2.8.A.1 An ideal spring has negligible mass and exerts a force that is proportional to the change in its length as measured from its relaxed length. A nonideal spring either has nonnegligible mass or exerts a force that is not proportional to the change in its length as measured from its relaxed length.
• 2.8.A.2 The magnitude of the force exerted by an ideal spring on an object is given by Hooke's law:
  • Equation: $\vec{F}_s = -k \Delta \vec{x}$
• 2.8.A.3 The force exerted on an object by a spring is always directed toward the equilibrium position of the object–spring system.

Learning Objective 2.8.B: Describe the equivalent spring constant of a combination of springs exerting forces on an object.

• 2.8.B.1 A collection of springs that exert forces on an object may behave as though they were a single spring with an equivalent spring constant $k_{\text{eq}}$.
  • 2.8.B.1.i The inverse of the equivalent spring constant of a set of springs in series is equal to the sum of the inverses of the individual spring constants.
    • Derived equation: $\dfrac{1}{k_{\text{eq, series}}} = \sum_i \dfrac{1}{k_i} = \dfrac{1}{k_1} + \dfrac{1}{k_2} + \dots$
  • 2.8.B.1.ii The equivalent spring constant of a set of springs arranged in series is smaller than the smallest constituent spring constant.
  • 2.8.B.1.iii The equivalent spring constant of a set of springs arranged in parallel is the sum of the individual spring constants.
    • Derived equation: $k_{\text{eq, parallel}} = \sum_i k_i = k_1 + k_2 + \dots$

Boundary statement: AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel.

Learning Objective 2.9.A: Describe the motion of an object subject to a resistive force.

• 2.9.A.1 A resistive force is defined as a velocity-dependent force in the opposite direction of an object's velocity, for example:
  • Equation: $\vec{F}_r = -k\vec{v}$
• 2.9.A.2 Applying Newton's second law to an object upon which a resistive force is exerted results in a differential equation for velocity.
  • 2.9.A.2.i Using the method of separation of variables, the velocity can be determined by integrating over the proper limits of integration.
  • 2.9.A.2.ii The acceleration or position of a moving object that is subject to a velocity-dependent force may be determined using initial conditions of the object and methods of calculus, once a function for velocity is determined.
  • 2.9.A.2.iii The position, velocity, and acceleration as functions of time of an object under the influence of a resistive force of the form $\vec{F}_r = -k\vec{v}$ are exponential and have asymptotes that are determined by the initial conditions of the object and the forces exerted on the object.
• 2.9.A.3 Terminal velocity is defined as the maximum speed achieved by an object moving under the influence of a constant force and a resistive force that are exerted on the object in opposite directions. The terminal condition is reached when the net force exerted on the object is zero.

Learning Objective 2.10.A: Describe the motion of an object traveling in a circular path.

• 2.10.A.1 Centripetal acceleration is the component of an object's acceleration directed toward the center of the object's circular path.
  • 2.10.A.1.i The magnitude of centripetal acceleration for an object moving in a circular path is the ratio of the object's tangential speed squared to the radius of the circular path.
    • Equation: $a_c = \dfrac{v^2}{r}$
  • 2.10.A.1.ii Centripetal acceleration is directed toward the center of an object's circular path.
• 2.10.A.2 Centripetal acceleration can result from a single force, more than one force, or components of forces that are exerted on an object in circular motion.
  • 2.10.A.2.i At the top of a vertical, circular loop, an object requires a minimum speed to maintain circular motion. At this point, and with this minimum velocity, the gravitational force is the only force that causes the centripetal acceleration.
    • Derived equation: $v = \sqrt{gr}$
  • 2.10.A.2.ii Components of the static friction force and the normal force can contribute to the net force producing centripetal acceleration of an object traveling in a circle on a banked surface.
  • 2.10.A.2.iii A component of tension contributes to the net force producing centripetal acceleration experienced by a conical pendulum.
• 2.10.A.3 Tangential acceleration is the rate at which an object's speed changes and is directed tangent to the object's circular path.
• 2.10.A.4 The net acceleration of an object moving in a circle is the vector sum of the centripetal acceleration and tangential acceleration.
• 2.10.A.5 The revolution of an object traveling in a circular path at a constant speed (uniform circular motion) can be described using period and frequency.
  • 2.10.A.5.i The time to complete one full circular path, one full rotation, or a full cycle of oscillatory motion is defined as period, $T$.
  • 2.10.A.5.ii The rate at which an object is completing revolutions is defined as frequency, $f$.
    • Equation: $T = \dfrac{1}{f}$
  • 2.10.A.5.iii For an object traveling at a constant speed in a circular path, the period is given by the derived equation
    • Derived equation: $T = \dfrac{2\pi r}{v}$

Learning Objective 2.10.B: Describe circular orbits using Kepler's third law.

• 2.10.B.1 For a satellite in circular orbit around a central body, the satellite's centripetal acceleration is caused only by gravitational attraction. The period and radius of the circular orbit are related to the mass of the central body.
  • Derived equation: $T^2 = \dfrac{4\pi^2}{GM} R^3$

Boundary statement: AP Physics C: Mechanics does not expect students to know Kepler's first or second laws of planetary motion.