Unit 2: 44 practice problems

(a) Calculate the system’s center-of-mass coordinates. Replace the system by a single object at a specific location with a specified mass. Justify when constituent properties or interactions can be ignored for this replacement, and name a question it cannot answer.

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(b) At the initial instant compare the individual motions of the loose, rigidly connected, and spring-connected modules. Explain why their internal structures change the needed analysis even though the total mass and external pull are identical. Calculate the center-of-mass acceleration and say which parts of the comparison the point model captures.

2.1.A.5

(a) Identify the interaction with the environment that transfers energy into the fixed cart system and calculate that transfer. Identify the mass transfer across the same boundary and find the system mass after the sand enters. Explain why choosing a boundary does not by itself make a system closed to either transfer.

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(b) Calculate the spring extension under each pair of pulls. Describe which external variable changes, which internal substructure changes, and why this change can occur while the system’s net external force remains zero.

2.1.A.6

(a) Explain why the person’s internal push cannot exert a net force on the person-plus-cart system or change its center-of-mass motion by itself. Identify the two recipient objects for the internal force pair and explain why relative motion is still possible.

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(b) Explain the microscopic physical origin of the person–wall push and the track’s normal contact force. State why these are contact interactions and why their electric origin does not make them additional electric-force arrows on a macroscopic free-body diagram.

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(c) Draw the whole-system free-body diagram after the rope is attached, with each actual external force as a straight arrow from the dot. Identify every source and direction, exclude internal forces, and calculate the horizontal center-of-mass acceleration.

2.2.A.1.ii

(a) State both ideal-string assumptions. State all ideal-pulley assumptions, explicitly including the axle’s location relative to the pulley’s center of mass. Explain which assumptions support treating the tension as the same across the pulley and the endpoints’ acceleration magnitudes as equal.

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(b) Explain how the external loading is transmitted by forces between successive string segments. Compare the tensions at the two ends, the two interior straight sections, and the portion over the ideal pulley. Explain why uniform interior tension would fail for a significantly massive hanging chain.

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(c) Draw the two masses’ separate free-body diagrams using actual force arrows, determine which mass accelerates downward, and calculate acceleration and the tension at every point of the ideal string.

2.3.A.3.ii

(a) Find the puck’s positions and interval velocities in observer A’s frame. Define an inertial reference frame in terms of an observer verifying Newton’s first law, and explain whether observer A’s result is consistent with that definition.

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(b) Find the puck’s positions and interval velocities in B’s frame. Explain whether B verifies Newton’s first law for this force-free puck, and why the apparent velocity change does not show a new horizontal contact force on the puck.

2.4.A.5

(a) Describe the gravitational interaction between the planet and either test object. Draw the attractive force pair along the line connecting their centers of mass, locate each modeled force’s application point, and compare the two forces’ magnitudes and directions. State the body-model condition that makes this center-line representation appropriate.

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(b) Use the defining ratio of gravitational force to test-object mass to find field strength in each trial. Independently calculate GM/r² and compare. Explain why the field does not grow when the test mass triples.

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(c) If gravity is the only force, find each object’s acceleration magnitude and direction. Explicitly compare its numerical value in m/s² with the field value in N/kg. Explain how the comparison changes if a support balances gravity.

2.6.A.2.ii

(a) Calculate the gravitational weight and scale reading in each block case. Evaluate the statement “any acceleration makes apparent weight different from gravitational weight.” Give the force-balance qualification needed for that statement and a numerical counterexample.

2.6.C.2

(b) Predict the test object’s local support-force reading in both sealed laboratories. Explain, using the equivalence principle, why an observer in the accelerating noninertial laboratory cannot distinguish its local apparent-weight effect from a uniform gravitational effect using only that measurement. State one limit of the comparison.

2.6.C.4

(a) Describe inertial mass and gravitational mass. Explain how inertial mass measures resistance to a change in motion under interaction and how gravitational mass relates to attraction between systems with mass. For a fixed net force, compare the acceleration of a larger inertial mass with a smaller one; for a fixed field, compare their gravitational-force responses if gravitational masses differ.

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(b) Calculate both mass values and use the stated uncertainties to assess agreement. State whether inertial and gravitational mass equivalence is a definition or an experimentally verified result. Describe one systematic effect that would need checking before interpreting a disagreement.

2.6.D.3

(a) Find the box surface’s velocity relative to the belt surface and the belt surface’s velocity relative to the box. Determine and justify the kinetic-friction direction on each surface, explicitly referring to their relative motion rather than ground velocity.

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(b) Draw the box’s actual force arrows and identify the normal force as the perpendicular contact component directed away from the surface. Calculate N, the kinetic-friction magnitude, and the box’s instantaneous acceleration. Describe how the normal-force direction would change if the belt were tilted.

2.7.A.2.ii

(a) Compare the kinetic-friction magnitudes for the two face orientations on the original material pair. Calculate them and explain what this equal-load comparison establishes about contact area in the ideal model. State why material and normal load must be controlled.

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(b) Calculate the kinetic-friction magnitude on the replacement surface. Explain what property of the contacting surfaces changes the coefficient and why the area result in part a does not imply all material pairs have the same friction.

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(c) For a typical dry material pair, state which is usually greater, μ_s or μ_k. Use the supplied original-pair values to compare maximum static friction with kinetic friction. Explain why this does not make the actual static force automatically equal to its maximum.

2.7.B.3

(a) State the ideal-spring mass assumption and its proportional force–length-change relation. Calculate k and predict the holding-force magnitude at total spring length 0.21 m. Explicitly identify the relaxed-length reference and explain why total length cannot replace deformation.

2.8.A.1

(b) Derive the hanging equilibrium extension. When the mass is 0.010 m above that equilibrium, calculate the spring force and net force with signs and directions. Evaluate the claim that the spring force always points toward the object–spring system’s equilibrium; identify the correctly scoped horizontal case and explain the counterexample.

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(c) Sketch spring force and net force versus y for the hanging system, labeling slopes and intercepts. Explain why their intercepts differ even though both lines have slope −k, and identify which force is zero at system equilibrium.

2.8.A.3

(a) Draw the bead’s velocity, centripetal-acceleration, and tangential-acceleration directions at the stated point, separately from any force diagram. Obtain tangential acceleration as the rate of speed change and calculate centripetal acceleration and total acceleration magnitude. Explain which component changes speed and which changes direction.

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(b) If instead the bead slows at the same speed-change magnitude while moving counterclockwise at that point, describe the new tangential and centripetal directions. Then state which acceleration component vanishes for constant speed and why the bead would still accelerate.

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(c) At the top of the vertical track, derive and calculate the minimum local speed that maintains contact. Explain which force causes centripetal acceleration at this threshold. Test whether a top speed of 4.0 m/s could maintain contact using the normal-force sign; do not infer a bottom launch speed.

2.9.A.2.i

(a) Draw the actual forces for a car at the frictionless design speed and describe the normal force’s inward and vertical components in a separate component construction. Explain which component provides centripetal acceleration and why normal force is not automatically equal to weight.

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(b) For speeds below and above the frictionless design speed, predict the tendency to slide and the static-friction direction along the bank. For each case explain whether its horizontal component adds to or subtracts from the normal force’s inward component and how its vertical component changes the vertical force balance. State why actual slip cannot be excluded at arbitrarily high or low speeds.

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(a) Draw the actual force diagram and identify the component of tension that contributes to the net force producing centripetal acceleration. State the vertical and inward equations, and explain why no separate “centripetal force” interaction is added to the diagram.

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(b) Calculate circle radius, tension, inward net-force magnitude, speed, and period. Explain why using string length as the circle radius would give the wrong result.

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(c) At the same string length and angle, predict how doubling the mass changes tension, centripetal acceleration, and speed. Derive the cancellations that support your prediction.

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