Forces and free body diagrams
Read Tipler and Mosca (2004) chapter 4 (Newton’s
Laws) and chapter 5 (Application of Newton’s laws); Pelcovits and Farkas (2024)
chapter 3 (Newton’s laws). Skim Tipler and Mosca (2004) chapter 11 (Gravity)
and chapter 12 (Static equilibrium and elasticity) and Pelcovits and Farkas (2024)
chapter 9 (Universal gravitation).
Gravitational force
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 \[\begin{equation} | \vec{F}_G | = G \dfrac{m_1 m_2}{r^2}. \end{equation}\]
The gravitational force is attractive. The gravitational force is always exerted along the line connecting the center of mass of the two interacting systems. The gravitational force on a system can be considered to be exerted on the system’s center of mass
Gravitational field
A gravitational field models the effects of a noncontact force exerted on an object at various positions in space.
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. \[\begin{equation} | \vec{g} | = \dfrac{| \vec{F}_g|}{m} = G\dfrac{M}{r^2} \end{equation}\] If the gravitational force is the only force exerted on an object, the observed acceleration of the object (in \(\unit{\meter\per\second\squared}\)) is numerically equal to the magnitude of the gravitational field strength (in \(\unit{\newton\per\kilo\gram}\)) at that location.
The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight. \[\begin{equation} \text{weight} = F_g = mg \end{equation}\]
Situations in which the gravitational force can be considered constant
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. Near the surface of Earth, the strength of the gravitational field is \[\begin{equation} g = \dfrac{G M_E}{R_E^2} = \qty{9.8}{\meter\per\second\squared} \sim \qty{10}{\meter\per\second\squared} \end{equation}\]
Conditions under which the magnitude of a system’s apparent weight is different from the magnitude of the gravitational force exerted on that system
The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system. 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. 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.
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.
Inertial and gravitational mass
Objects have inertial mass, or inertia, a property that determines how much an object’s motion resists changes when interacting with another object. Gravitational mass is related to the force of attraction between two systems with mass. Inertial mass and gravitational mass have been experimentally verified to be equivalent.
Gravitational force exerted on an object by a uniform spherical distribution of mass
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. AP Physics C: Mechanics does not expect students to
mathematically prove or derive Newton’s shell theorem
Newton’s shell theorem describes the net gravitational force exerted on an object by a uniform spherical shell of mass. The net gravitational force exerted on an object inside a thin spherical shell is zero. 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.
An object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere. 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. \[\begin{equation} m_{partial} = \rho \dfrac{4}{3} \pi r_{partial}^2 \end{equation}\]
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. \[\begin{equation} F_{g,partial} = -k r_{partial} \end{equation}\]
Kinetic and static friction
Kinetic friction
Kinetic friction occurs when two surfaces in contact move relative to each other. The kinetic friction force is exerted in a direction opposite the motion of each surface relative to the other surface. The force of friction between two surfaces does not depend on the size of the surface area of contact.
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: \[\begin{equation} F_{fr,k} = \mu_k N \end{equation}\]
The coefficient of kinetic friction depends on the material properties of the surfaces that are in contact. 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.
Static friction
Static friction may occur between the contacting surfaces of two objects that are not moving relative to each other. Static friction adopts the value and direction required to prevent an object from slipping or sliding on a surface. \[\begin{equation} F_{fr,s} \leq \mu_s N \end{equation}\] Slipping and sliding refer to situations in which two surfaces are moving relative to each other. There exists a maximum value for which static friction will prevent an object from slipping on a given surface. \[\begin{equation} F_{fr,s,max} = \mu_s N \end{equation}\]
The coefficient of static friction is typically greater than the coefficient of kinetic friction for a given pair of surfaces.
Spring forces
Ideal spring
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. The magnitude of the force exerted by an ideal spring on an object is given by Hooke’s law: \[\begin{equation} \vec{F}_s = -k \Delta \vec{x} \end{equation}\] The force exerted on an object by a spring is always directed toward the equilibrium position of the object–spring system.
Equivalent spring constant of a combination of springs
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.
A collection of springs that exert forces on an object may behave as though they were a single spring with an equivalent spring constant \(k_{eq}\).
For springs in series, 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: \[\begin{equation} \dfrac{1}{k_{eq}} = \Sigma_i \dfrac{1}{k_i} = \dfrac{1}{k_1} + \dfrac{1}{k_2} + ... \end{equation}\] The equivalent spring constant of a set of springs arranged in series is smaller than the smallest constituent spring constant.
For springs in parallel, the equivalent spring constant of a set of springs arranged in parallel is the sum of the individual spring constants: \[\begin{equation} k_{eq} = \Sigma_i k_i = k_1 + k_2 + ... \end{equation}\]
Resistive forces
A resistive force is defined as a velocity dependent force in the opposite direction of an object’s velocity, for example: \[\begin{equation} \vec{F}_r = -k \vec{v} \end{equation}\] An example of a resistive force of this form might be viscous friction, due to a lubrication film being sheared as one object slides relative to another, such as in a bearing.
Applying Newton’s second law to an object upon which a resistive force is exerted results in a differential equation for velocity. Using the method of separation of variables, the velocity can be determined by integrating over the proper limits of integration. 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.
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.
Terminal velocity is defined as the maximum speed achieved by an object moving under the influence of a constance 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. An example of this is a small sphere falling through a viscous liquid (ignoring buoyancy): \[\begin{align} -k v - mg &= m\dfrac{dv}{dt} \\ -k v_T - mg &= 0 \\ v_T = - \dfrac{mg}{k} \end{align}\]
Drag (optional)
At higher Reynolds number, a more common form of resistive force is given by drag: \[\begin{equation} D = C_D \frac{1}{2} \rho u^2 A \end{equation}\] where \(C_D\) is a drag coefficient, \(\rho\) is the fluid density, \(u\) is velocity, and \(A\) is a reference area. Drag arises from skin friction and shear stresses as well as form drag, in which the shape and streamlining (or lack thereof) result in wakes with eddies and dead zones and turbulence. The latter lead to low pressure behind the object compared to high stagnation pressure in the front of the object, resulting in a force that resists forward motion. You are not responsible for this form of resistive force on the AP Physics C Mechanics test.
Circular motion
Motion of an object traveling in a circular path
Centripetal acceleration is the component of an object’s acceleration directed toward the center of the object’s circular path.
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. \[\begin{equation} a_c = \dfrac{v^2}{r} \end{equation}\] Centripetal acceleration is directed toward the center of an object’s circular path.
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.
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. \[\begin{equation} v = \sqrt{g r} \end{equation}\]
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.
A component of tension contributes to the net force producing centripetal acceleration experienced by a conical pendulum.
Tangential acceleration is the rate at which an object’s speed changes and is directed tangent to the object’s circular path.
The net acceleration of an object moving in a circle is the vector sum of the centripetal acceleration and tangential acceleration.
The revolution of an object traveling in a circular path at a constant speed (uniform circular motion) can be described using period and frequency.
The time to complete one full circular path, one full rotation, or a full cycle of oscillatory motion is defined as period, \(T\)
The rate at which an object is completing revolutions is defined as frequency, \(f\). \[\begin{equation} T = \dfrac{1}{f} \end{equation}\]
For an object traveling at a constant speed in a circular path, the period is given by the derived equation \[\begin{equation} T = \dfrac{2\pi r}{v} \end{equation}\]
Example: Kepler’s third law
Consider a satellite of mass \(m\) in a circular orbit of radius \(R\) around a central body of mass \(M\). The centripetal acceleration is given by \[a_c = \dfrac{v^2}{R} \] \[ = \dfrac{(\dfrac{2\pi R}{T})^2}{R} \] \[ = \dfrac{4 \pi^2 R}{T^2}.\qquad{(1)}\]
The sum of the forces is \[\begin{equation} \Sigma F = \dfrac{G M m}{R^2} \end{equation}\qquad{(2)}\]
Combining eq. 1 and eq. 2 and rearranging gives \[\begin{align} \dfrac{G M m}{R^2} &= m \dfrac{4 \pi^2 R}{T^2} \\ T^2 &= \dfrac{4 \pi^2}{G M} R^3. \end{align}\]
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. \[\begin{equation} T^2 = \dfrac{4\pi^2}{GM} R^3 \end{equation}\]
See also
- list here