Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions mix the essential knowledge threads of Topic 2.1 — how a system's properties emerge from the interactions between its parts and when it can be simplified to a single object, how symmetry locates a center of mass, the discrete weighted-average formula for point masses, the continuous center-of-mass integral and linear mass density, finding total mass by integrating a density function over length, area, or volume, and modeling an entire system as a single object at its center of mass.
A block on a table is connected by a string, over a pulley, to a mass hanging off the table's edge. If the system is defined as the block and the hanging mass together, how should the string tension be classified?
Select A, B, C, or D to check your answer.
Explain why it's reasonable to model an entire car as a single object when finding its overall acceleration leaving a stoplight, but why that same simplification fails if you instead need the force the seatbelt exerts on a passenger during that acceleration.
A meterstick has uniform density along its entire 100 cm length. At which position is its center of mass located?
Select A, B, C, or D to check your answer.
A uniform disk has a circular hole cut out of its right side, off-center. Explain whether the disk's center of mass remains at its original geometric center, and if not, which way it shifts and why.
A rod's linear density increases steadily from one end to the other, even though the rod's shape is geometrically symmetric about its midpoint. Can the symmetry shortcut (center of mass lies on a line of symmetry) be used to find this rod's center of mass?
Select A, B, C, or D to check your answer.
A 2 kg point mass sits at x = 0 m and a 4 kg point mass sits at x = 3 m. Find the center of mass of this two-mass system, and explain why your answer lands closer to one mass than the other.
Point masses of 1 kg, 2 kg, and 3 kg sit at x = 0 m, x = 2 m, and x = 5 m, respectively. What is the system's center of mass?
Select A, B, C, or D to check your answer.
A 2 kg mass sits at x = 0 m and a 3 kg mass sits at x = 4 m. Where should a 5 kg mass be placed so that the center of mass of all three masses ends up at exactly x = 2 m?
If every mass in a discrete system is doubled, while each mass's position stays exactly the same, what happens to the system's center of mass?
Select A, B, C, or D to check your answer.
A rod's mass as a function of position is m(x) = cx², where c is a constant. Find the rod's linear mass density λ(x), and explain what it means physically that λ increases with x.
A rod of length L has uniform linear density λ = M/L. Evaluating the center-of-mass integral xcm = ∫xλ dx / ∫λ dx over the rod gives which result?
Select A, B, C, or D to check your answer.
A rod of length L has linear density λ(x) = λ0(x/L), measured from one end. Find the rod's total mass and its center of mass.
For a rod whose density increases steadily from one end to the other, where does the center of mass lie relative to the rod's geometric midpoint?
Select A, B, C, or D to check your answer.
A rod from x = 0 to x = L = 2 m has linear density λ(x) = 3x² (in kg/m, x in meters). Find the rod's total mass.
A uniform disk of radius r = 2 m has a constant areal (surface) density σ = 5 kg/m². What is the disk's total mass?
Select A, B, C, or D to check your answer.
A solid sphere of radius R has volumetric density ρ(r) = ρ0(r/R), varying with radial distance r. Using a thin spherical shell of thickness dr (volume dV = 4πr² dr) as the mass element, find the sphere's total mass.
Which expression correctly gives the total mass of a solid whose density varies throughout its volume?
Select A, B, C, or D to check your answer.
Explain why a high jumper's center of mass follows a simple parabolic path through the air during a jump, even though the jumper's arms and legs move in complicated, independent ways.
Two carts on a frictionless track are connected by a spring and released from rest. As the carts oscillate back and forth, what happens to the center of mass of the two-cart system?
Select A, B, C, or D to check your answer.
For the same two-cart-and-spring system, explain why the single-object (center-of-mass) model cannot be used to find the force the spring exerts on one of the individual carts, even though it correctly describes the system's overall motion.