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Please find my solutions to past F=ma Contest problems below. Past Exams Problems organized by topic Quarter-final Exams Problems organized by topic
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Topic: CollisionsConcepts: 1D elastic collision, 1D inelastic collision, Limiting cases Solution: We study the two limiting cases of a perfectly inelastic collision and an elastic collision which will provide us with the bounds that apply to any generic collision. For a perfectly inelastic collision, the masses stick together so For an elastic collision, recall Thus,
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Topic: FluidsConcepts: Buoyant force, Equivalence principle, Floating Solution: We initially balance the weight of the block with the buoyant force, so the fraction submerged is given by In the accelerating frame moving upward with aa, the effective gravity is g′=g+ag’=g+a since we include the fictitious inertial force −ma→-m\vec{a}. Thus, our force balance equation becomes which
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Topic: EnergyConcepts: Air resistance, Power Solution: To travel at velocity vv in the presence of air friction ff, we need power We are given P∝mP \propto m and f∝Av2f \propto Av^2 so It remains to express AA in terms of mm. For a fixed density, m∝L3m \propto L^3 where LL is the side length of
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Topic: DynamicsConcepts: Cutting strings vs. springs, Newton’s laws, Statics Solution: Initially when the blocks are at rest, the tension T1T_1 in the top string holds the weight of both blocks while the tension T2T_2 in the bottom string holds the weight of m2m_2 so If the top string is cut at the connection point to
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Topic: EnergyConcepts: Circular motion, Conservation of energy, Forces in mechanics Solution: If the bead leaves the sphere at angle θ\theta, we can find its speed by conservation of energy. Setting the potential energy reference at the center of the sphere, we have Since Ei=EfE_i=E_f, To find θ\theta, we note the bead is undergoing circular motion
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Topic: GravityConcepts: Conservation of angular momentum, Kepler’s laws Solution: Kepler’s laws: Thus, the answer is B.
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Topic: Oscillatory MotionConcepts: Forces in mechanics, Mass-spring system, Simple harmonic motion Solution: Suppose the springs have equilibrium length ll and spring constant kk. If we displace the mass mm by distance xx parallel to the spring, the spring force FF is given by Then the restoring force is For finite xx, the magnitude of this
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Topic: DynamicsConcepts: Equivalence principle, Projectile motion Solution: In the accelerating frame of the box, we introduce the (fictitious) inertial force −ma→-m\vec{a} on the particle. This has the same form as gravity so combining them results in an effective gravity geffg_{\text{eff}} directed in the lower left direction. The particle now behaves effectively as a projectile with
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Topic: DynamicsConcepts: Circular motion, Inclined plane, Limiting cases Solution: Looking top-down at the bead sliding down the helix, we see that it has a tangential acceleration ata_t determined by the pitch of the helix and a radial acceleration ar=v2/ra_r=v^2/r. The total acceleration is Since v=attv=a_tt, we have To determine the graph of a(t)a(t), we look
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Topic: EnergyConcepts: Work-energy theorem Solution: There is no net work done on the textbook since it starts at rest and ends at rest. Gravity does work while the snow does work Since Wtot=0W_{\text{tot}}=0, Solving for the retarding force FF, so the answer is E.
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Topic: FluidsConcepts: Buoyant force, Floating Solution: Before additional oil is added, we balance forces on the cylinder. The buoyant forces from the oil and water must equal the weight: Thus, the density of the object is Once the cylinder is only in oil, recall the ratio of densities yields the fraction of the object submerged
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Topic: Rigid BodiesConcepts: Angular kinematics, Rolling motion, Velocity constraints Solution: Since the car (rigid object) is traveling in a circle, all parts of the car move at the same angular velocity ωcar\omega_{\text{car}} around the center. If the left (right) wheel has velocity vlv_l (vrv_r) then we must have The velocity vv of the wheel is
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1D elastic collision 1D inelastic collision 5 kinematics equations Air resistance Angular kinematics Angular momentum Artificial gravity Atwood machine Average vs. instantaneous Ballistic pendulum Bernoulli's principle Buoyant force Center of mass Circular motion Circular orbits CM frame Conservation of angular momentum Conservation of energy Conservation of linear momentum Coriolis force Data expression Dimensional analysis Dynamics Dynamics of CM Effective spring constant Elliptical orbits Energy dissipation Equivalence principle Error propagation Escape velocity Fictitious forces Floating Fma: Collisions Fma: Dynamics Fma: Energy Fma: Fluids Fma: Gravity Fma: Kinematics Fma: Oscillatory Motion Fma: Other Fma: Rigid Bodies Fma: System of Masses Forces in mechanics Free fall Gauss's law for gravity General angular momentum General kinematics equations General kinetic energy General Newton's 2nd law Gravitational force Gravitational potential energy Impulse-momentum theorem Inclined plane Kepler's laws Kinematics Kinetic energy Law of reflection Limiting cases Mass-spring system Mechanics Moments of inertia Motion graphs Newton's laws Parallel-axis theorem Pascal's law Perpendicular-axis theorem Physical pendulum Potential energy graphs Power Pressure with depth Projectile motion Pulleys with rotational inertia Qtrfin: E&M Qtrfin: Mechanics Recursion for rotational inertia Reduced mass Relative velocity Rigid Bodies Rolling down inclined plane Rolling motion Rotational Newton's 2nd law Scale reading Simple harmonic motion Simple pendulum Small angle approximation Speed vs. velocity Spring potential energy Statics Surface tension Tensile strength Torque Torque from weight Types of equilibrium Vectors Velocity constraints Vertical spring Wave speed in string Work Work-energy theorem Young's modulus
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