AP Physics C: Mechanics Practice Quiz: Systems and Center of Mass
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 15 questions to check your progress.
Question 1 of 15
All Questions (15)
A) The interactions between objects within the system.
B) The total mass of the system.
C) The system's velocity relative to an external observer.
D) The external forces acting on the system.
Correct Answer: A
The provided content explicitly states, "System properties are determined by the interactions between objects within the system."
A) It is always located at the position of the most massive object.
B) It is a position determined by the distribution of mass of the system's constituent parts.
C) It is always located at the geometric center of the system.
D) It is the point where the net external force is applied.
Correct Answer: B
The content states that the center of mass location is described "with respect to the system's constituent parts," which means it is a weighted average based on the mass and position of those parts.
A) 3.5 m
B) 4.0 m
C) 5.0 m
D) 7.0 m
Correct Answer: B
Using the equation $\overline{x}_{cm}=\frac{\sum m_{i}\overline{x}_{i}}{\sum m_{i}}$, the calculation is $\overline{x}_{cm}=\frac{(2 \text{ kg})(1 \text{ m}) + (3 \text{ kg})(6 \text{ m})}{2 \text{ kg} + 3 \text{ kg}} = \frac{2 + 18}{5} = \frac{20}{5} = 4.0$ m.
A) exactly halfway between m₁ and m₂.
B) closer to the smaller mass, m₂.
C) closer to the larger mass, m₁.
D) at the location of the larger mass, m₁.
Correct Answer: C
The center of mass is a mass-weighted average of the positions. The more massive object has a greater influence on this average, pulling the center of mass closer to its own position.
A) The total mass of the body.
B) The total length of the body.
C) The linear mass density at a specific position.
D) The center of mass of the body.
Correct Answer: C
The provided content defines λ=(d/dl)m(l) as the linear mass density, which is the derivative of mass with respect to position, representing the mass per unit length at a point.
A) $\overline{x}_{cm}=\frac{\sum m_{i}\overline{x}_{i}}{\sum m_{i}}$
B) $\overline{r}_{cm}=\frac{\int\overline{r}dm}{\int dm}$
C) $\lambda=\frac{d}{dl}m(l)$
D) F = ma
Correct Answer: B
The rod is a nonuniform solid, which can be considered a collection of differential masses (dm). The equation $\overline{r}_{cm}=\frac{\int\overline{r}dm}{\int dm}$ is specifically provided for calculating the center of mass of such a continuous object.
A) 2.00 m
B) 2.67 m
C) 3.00 m
D) 6.00 m
Correct Answer: B
Using the formula $\overline{x}_{cm}=\frac{\sum m_{i}\overline{x}_{i}}{\sum m_{i}}$, we get $\overline{x}_{cm}=\frac{(1)(0) + (2)(2) + (3)(4)}{1+2+3} = \frac{0+4+12}{6} = \frac{16}{6} \approx 2.67$ m.
A) dm = λ dx
B) dm = λ / dx
C) dm = x dλ
D) dm = λ x
Correct Answer: A
The definition λ = dm/dl (or dm/dx for the x-axis) can be rearranged into its differential form, dm = λ dl (or dm = λ dx). This substitution is necessary to perform the integration with respect to position, x.
A) At the location of one of the masses.
B) Outside the rod, closer to one end.
C) At the geometric center of the rod connecting the masses.
D) The location cannot be determined without knowing the value of m.
Correct Answer: C
For a system of two equal masses, the center of mass is located exactly halfway between them. Since the rod connects them, this point is the geometric center of the rod.
A) The total volume of the solid.
B) The total mass of the solid.
C) The average density of the solid.
D) The moment of inertia of the solid.
Correct Answer: B
The integral ∫dm represents the sum of all the differential mass elements that make up the solid. This sum is, by definition, the total mass of the solid (M_total).
A) The system's properties are solely determined by the planet's mass.
B) The system's properties are determined by the gravitational interaction between the planet and the moon.
C) The center of mass must be located inside the planet.
D) The system has no interactions because it is in the vacuum of space.
Correct Answer: B
The content states that "System properties are determined by the interactions between objects within the system." For a planet-moon system, the primary interaction is gravity.
A) $\lambda=\frac{d}{dl}m(l)$
B) $\overline{r}_{cm}=\frac{\int\overline{r}dm}{\int dm}$
C) $\overline{x}_{cm}=\frac{\sum m_{i}\overline{x}_{i}}{\sum m_{i}}$
D) $x = v_0t + \frac{1}{2}at^2$
Correct Answer: C
This is a direct recall question. The content explicitly provides the equation $\overline{x}_{cm}=\frac{\sum m_{i}\overline{x}_{i}}{\sum m_{i}}$ for calculating the center of mass of a system of discrete parts along an axis.
A) The mass is concentrated at the center of the rod.
B) The mass is concentrated at the ends of the rod.
C) The mass per unit length is constant along the entire rod.
D) The total mass of the rod is equal to λ.
Correct Answer: C
Linear mass density, λ, is the mass per unit length. If it is uniform, it means this value is constant everywhere on the rod. The derivative dm/dl is a constant.
A) It moves in the positive x-direction.
B) It moves in the negative x-direction.
C) It does not move.
D) It moves to x = 0 m.
Correct Answer: A
The initial center of mass is $\overline{x}_{cm1} = \frac{(1)(-2) + (3)(2)}{1+3} = \frac{4}{4} = 1$ m. The final center of mass is $\overline{x}_{cm2} = \frac{(1)(0) + (3)(2)}{1+3} = \frac{6}{4} = 1.5$ m. The center of mass moves from 1 m to 1.5 m, which is a shift in the positive x-direction.
A) When the system consists of only two point masses.
B) When the system's mass is distributed continuously throughout a solid object.
C) When all objects in the system have equal mass.
D) When the system is one-dimensional.
Correct Answer: B
The summation formula is for discrete systems (a collection of separate point masses). The integral formula is the extension of this idea for continuous systems (like a solid rod or plate) where the mass is spread out and can be treated as an infinite number of differential mass elements.