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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: Oscillatory MotionConcepts: Effective spring constant, Mass-spring system Solution: When the two masses oscillate in phase, they always move together so we can equivalently replace them with a single particle of mass 2m2m. The two springs on the sides are connected in parallel so we can replace them with a single spring with effective constant
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Topic: OtherConcepts: Standing waves, Tensile strength, Wave speed in string Solution: Recall the wavelength of the fundamental mode is given by λ=2L\lambda=2L. The fundamental frequency ff is then Since the speed of a wave in a string is given by we have since we are considering strings of the same length. In terms of the
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Topic: System of MassesConcepts: 1D inelastic collision, Conservation of energy, Vertical spring Solution: Conserving energy for m1m_1, we have so m1m_1 hits m2m_2 with velocity v1=2ghv_1=\sqrt{2gh}. Conserving momentum for the perfectly inelastic collision, so both masses move with velocity after the collision. To find the maximum displacement xx of m2m_2 from its original location, we
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Topic: Rigid BodiesConcepts: Energy dissipation, Limiting cases, Rolling down inclined plane Solution: Recall the acceleration of an object rolling without slipping down an incline is given by where the object has moment of inertia I=βmr2I=\beta mr^2. For small enough θ\theta, static friction is large enough to provide sufficient torque since Using a=rαa=r\alpha as the object
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Topic: KinematicsConcepts: 1D inelastic collision, Free fall Solution: If we launch a ball upward with velocity vv, then it takes time to reach the top. Coming down takes the same amount of time so the time for the first bounce is Letting the coefficient of restitution be rr, the starting speed for the second bounce
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Topic: Oscillatory MotionConcepts: Newton’s laws, Simple harmonic motion Solution: Suppose we displace the water so that the right side moves up by distance xx and the left side moves down by distance xx. Then the extra weight on the right side is where AA is the cross-sectional area of the U-tube. This force accelerates all
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Topic: OtherConcepts: Data expression Solution: We can’t determine anything quantitative from the curves so we use the linear graph. Since logy\log y is linear in xx, we have letting a=ea2a=e^{a_2} and b=a1b=a_1. Thus, the answer is D.
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Topic: OtherConcepts: Dimensional analysis, Kinetic energy Solution: Let λ\lambda be the maximum kinetic energy per kilogram that can be stored in the flywheel. If we double the thickness hh of the flywheel, that is equivalent to stacking two flywheels on top of each other. Since each has λ\lambda energy per kilogram, the combination also has
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Topic: EnergyConcepts: Potential energy graphs Solution: Recall a particle moving in a potential energy landscape U(x)U(x) experiences a force Thus, the answer is E.
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Topic: DynamicsConcepts: Circular motion, Newton’s laws Solution: Looking at one of the masses, we see that the tension provides the centripetal acceleration, so the answer is C. Topic: System of MassesConcepts: Conservation of angular momentum, Work-energy theorem Solution: There is no net external torque on the system of two masses since the tensions pulling them
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Topic: DynamicsConcepts: Circular motion, Conservation of energy, Newton’s laws Solution: The pendulum feels the forces of tension and gravity. Applying Newton’s 2nd law in the tangential direction, Since the pendulum is undergoing circular motion, the radial acceleration is given by The magnitude of the total acceleration is Now we go through the possible answer choices.
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Topic: FluidsConcepts: Floating Solution: Recall the fraction of an object that is submerged in a fluid is given by In our case, we have The fraction of the object that would be submerged in oil is using the fact that ρoil=34ρwater\rho_{\text{oil}}=\frac{3}{4}\rho_{\text{water}}. Thus, the answer is C.
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Topic: CollisionsConcepts: Conservation of linear momentum, Vectors Solution: By conservation of linear momentum, Since p→1\vec{p}_1 makes a right angle with p→2\vec{p}_2, the three momenta vectors form the sides of a right triangle. Thus, Solving for v1fv_{1f}, so the answer is A. Topic: EnergyConcepts: Kinetic energy Solution: The initial kinetic energy is given by We found
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Topic: CollisionsConcepts: 1D inelastic collision Solution: The initial momentum of the system is Since the collisions are completely inelastic, all the masses stick together so the final momentum is Conserving linear momentum pi=pfp_i=p_f, so the answer is A. Topic: CollisionsConcepts: 1D elastic collision Solution: Recall that in an elastic collision between mass mm moving at
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Topic: CollisionsConcepts: Conservation of linear momentum, Dynamics of CM Solution: Since there is no net external force on the system, we have conservation of linear momentum ptotp_{\text{tot}}. Recall that Since linear momentum is conserved, we can compute it at the beginning, We have Mtot=m1+m2+m3M_{\text{tot}}=m_1+m_2+m_3 so Thus, the answer is A.
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Topic: KinematicsConcepts: Projectile motion, Vectors Solution: Recall for projectile motion, we have where the first term describes straight line motion if the particle was free and the second term accounts for the deflection by gravity. Since the distance between the ball and target is d=150md=150\,\mathrm{m}, the time taken to reach the wall is The deflection
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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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