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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: KinematicsConcepts: Free fall Solution: From kinematics, we have where hh is the height an object is dropped and tt is the time it takes to reach the ground. Suppose the first ball above the ground at height h1h_1 takes time TT to hit the floor. We want to find the height of the nnth
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Topic: Rigid BodiesConcepts: Statics, Torque Solution: We choose the pivot point at the contact point between the disk and the ground so that gravity, normal force, and friction don’t contribute to the torque. Then, the critical angle between rolling left (at smaller angles) and rolling right (at larger angles) is where the tension force goes
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Topic: GravityConcepts: Gravitational force, Statics Solution: For the case of two balls, each mass feels the gravitational force FF from the other mass and the compressive force from the rod. To balance gravity, the compressive force also has magnitude FF. For the case of three balls, each mass feels the gravitational forces from the other
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Topic: FluidsConcepts: Buoyant force Solution: Initially, we have the metal cup floating in the tub with height difference hh between the water level outside and inside the cup (the bottom of the cup before the tap is turned on). After the tap is turned on, the height difference between the water levels outside and inside
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Topic: Oscillatory MotionConcepts: Equivalence principle, Simple pendulum, Small angle approximation Solution: Recall the period of a simple pendulum has the form (by dimensional analysis): Thus, the answer is C.
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Topic: DynamicsConcepts: Forces in mechanics Solution: Note that pulling on both ends of the spring is equivalent to the usual scenario of pulling on the spring in one direction with the other end attached to a wall since the wall exerts the same force to keep the spring at rest. From the definition of the
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Topic: FluidsConcepts: Buoyant force, Floating Solution: We are given yielding the ratio Recall the fraction of an object that floats is given by In our case, we have so the answer is D.
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Topic: Rigid BodiesConcepts: Conservation of angular momentum, Kinetic energy Solution: Recall the rotational kinetic energy is given by Using the angular momentum L=IωL=I\omega, we can rewrite In our case, we have conservation of angular momentum since there is no external torque. Since LL is constant, we see that doubling the angular velocity also doubles the
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Topic: CollisionsConcepts: 1D inelastic collision Solution: By conservation of linear momentum, we have so the answer is E.
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Topic: GravityConcepts: Circular motion Solution: We know the period TT of the Earth’s rotation around the sun is 11 year. Assuming the Earth travels in uniform circular motion, we have so the answer is B.
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Topic: KinematicsConcepts: Motion graphs Solution: Recall the average acceleration is given by which on a vv–tt plot corresponds to the slope of the line connecting the beginning and end points. In our case, the magnitudes of the slopes are equal for all three objects so the answer is E. Topic: KinematicsConcepts: Motion graphs Solution: Recall
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Topic: KinematicsConcepts: Average vs. instantaneous Solution: Let tt be the time the cyclist traveled at 22km/hr22\,\mathrm{km/hr}. Then by definition of average speed, Solving for the time, Thus, so the answer is A.
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Topic: GravityConcepts: Circular orbits, Elliptical orbits Solution: Recall from the elliptical orbit equations that In our case, we have Adding the equations, Multiplying the equations, Substituting into the expression for vapogeev_{\text{apogee}}, The velocity of a circular orbit of radius 2R2R is so the answer is A.
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Topic: Rigid BodiesConcepts: Moments of inertia, Parallel-axis theorem Solution: The moment of inertia of the original disk is To calculate the moment of inertia of the removed disk, note that since M∝R2M \propto R^2, it has radius r=R/2r=R/2 and mass m=M/4m=M/4. Its moment of inertia around the axis (through its edge) is where we used
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Topic: FluidsConcepts: General Newton’s 2nd law, Statics Solution: First, we study the force on one U-tube: To find the force the stream of water exerts on the U-tube, we consider a small time interval Δt\Delta t where a small mass Δm\Delta m of water enters from the left at the top of the U-tube. Since
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Topic: DynamicsConcepts: Equivalence principle, Fictitious forces Solution: In the noninertial frame of the car, we add a centrifugal force directed outwards from the center of the circle. Combining the fictitious force with gravity, We find the effective gravity points in the lower left direction. The balloon floats in the opposite direction so the answer is
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Topic: GravityConcepts: Binding energy, Dimensional analysis Solution: Recall the binding energy of a solid sphere of mass MM and radius RR is We could also have obtained the scaling from dimensional analysis. We have If the radius is doubled, the energy increases by a factor of 25=322^5=32 so the answer is E.
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Topic: EnergyConcepts: Potential energy graphs Solution: Recall force is related to the potential energy by so force is given by the negative slope of the UU–xx plot. Part I and Part IV have zero slope so there is no force. Part II has slope −1-1 so the force is +1N+1\,\mathrm{N}. Part II has slope +1+1
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Topic: GravityConcepts: Gravitational potential energy Solution: Each pair of masses contributes gravitational potential energy to the arrangement. We have (42)=6{4\choose 2}=6 pairs so The answer is D.
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Topic: System of MassesConcepts: Conservation of energy, Conservation of linear momentum Solution: When the block reaches its maximum height, it is at rest with respect to the incline. Thus, they both have the same velocity vv. Conserving linear momentum, To find the height hh, we conserve energy, so the answer is C. Topic: System of
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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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