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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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Solution: B Concepts:
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Solution: B Concepts:
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Solution: D Concepts:
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Topic: OtherConcepts: Error propagation Solution: Recall independent errors add in quadrature: Alice: One measurement with uncertainty ΔL2=2mm\Delta L_2=2\,\mathrm{mm}. Bob: Two measurements with uncertainty ΔL2=2mm\Delta L_2=2\,\mathrm{mm} added together. Christina: Two measurements with uncertainty ΔL1=1mm\Delta L_1=1\,\mathrm{mm} added together. Thus, so the answer is A.
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Topic: Rigid BodiesConcepts: Moments of inertia, Parallel-axis theorem Solution: We are given that the moment of inertia I1I_1 of an equilateral triangle (axis through side) is Let’s first find the moment of inertia I2I_2 of an equilateral triangle (axis through vertex). We use the parallel axis theorem: Subtracting these two equations, The hexagon can be…
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Topic: DynamicsConcepts: Circular motion, Forces in mechanics, Small angle approximation Solution: We analyze a small piece of the rubber band with mass dMdM that subtends an angle dθd\theta. The spring forces TT on its ends provide the centripetal acceleration. We have Using the small-angle approximation sinθ≈θ\sin\theta \approx \theta, Since the rubber band has uniform density,…
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Topic: System of MassesConcepts: Center of mass, Dynamics of CM Solution: There are no external forces on the system in the horizontal direction. Thus, the CM does not move in the horizontal direction. Setting the final position of mm as the origin, the CM is located a distance to the left. Thus, the answer is…
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Topic: Rigid BodiesConcepts: Center of mass, Inclined plane, Statics, Torque from weight Solution: Recall the slip angle on an incline is given by where μ\mu is the coefficient of friction. To find the tipping angle, note that when the triangle is about to topple over, the normal force and friction only act at the bottom…
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Topic: EnergyConcepts: Conservation of energy, Projectile motion Solution: By conservation of energy, when the bead travels to the bottom of the semicircle, it has speed From kinematics, it takes time to fall a distance HH. The range DD is given by Since D2D^2 and RHRH are proportional to each other, their graph is a line…
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Topic: Rigid BodiesConcepts: 5 kinematics equations, Angular kinematics Solution: Let t1t_1 be the time it takes the object to accelerate with α\alpha to ω0\omega_0 from rest. Let t2t_2 be the time the object stays rotating at ω0\omega_0. Then the time it takes the object to go back to rest is also t1t_1 since it is…
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Topic: Rigid BodiesConcepts: Conservation of angular momentum, General angular momentum Solution: Choosing a point on the ground as the pivot point, note that there is no net torque on the ping pong ball. The line of the friction force goes through the pivot point and the normal force cancels out gravity. Thus, we have conservation…
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Topic: KinematicsConcepts: Projectile motion, Relative velocity Solution: We go to the truck frame where it is at rest and the ball is moving at v=8m/sv=8\,\mathrm{m/s} to the left. To drop by distance hh, it takes time In this time, the ball travels a distance to the left so the answer is C.
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Topic: Oscillatory MotionConcepts: Rotational Newton’s 2nd law, Simple harmonic motion, Small angle approximation Solution: We first find the original oscillation frequency ff. If we displace the rod by a small angle θ\theta, then the spring gets stretched by x=lθx=l\theta. The spring force FF is and the corresponding torque τ\tau is where we used the small…
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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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