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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: EnergyConcepts: Kinetic energy, Power Solution: Since there is no energy loss, all power from the engine goes to kinetic energy, The instantaneous power is given by Substituting our expression for the velocity, If the time is doubled, the acceleration decreases by a factor of 2\sqrt{2}, so the answer is B.
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Topic: GravityConcepts: Conservation of energy, Gravitational potential energy Solution: Recall the gravitational potential energy between two particles of masses mm and MM separated by distance rr is In our case, the particle starts very far away so there is no potential energy in the beginning, The particle maximizes its speed when it minimizes its potential
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Topic: DynamicsConcepts: Fictitious forces Solution: We can go to the noninertial box frame by introducing a fictitious −ma→-m\vec{a} force on the mass. This stretches the spring by distance which is also how much closer the mass moves toward the bottom. Thus, the answer is E.
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Topic: DynamicsConcepts: Forces in mechanics Solution: Applying Newton’s 2nd law to the ball when it is at height yy above the original position of the spring, taking downwards to be negative as indicated. We have so the answer is E.
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Topic: DynamicsConcepts: Statics, Torque from weight Solution: We balance torques around the bottom of the right leg. When the table is about to tip over, the normal force from the left leg goes to zero. Therefore, the only contributions are from the weights of the table and carpenter. We have so the answer is D.
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Topic: System of MassesConcepts: Conservation of angular momentum, Kinetic energy Solution: Since there are no external torques on the system, we have conservation of angular momentum LL. Initially, the device has rotational kinetic energy If the angular velocity is doubled, then the energy is also doubled, so the answer is B.
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Topic: Oscillatory MotionConcepts: Conservation of energy, Spring potential energy Solution: By conservation of energy, If kk is increased by a factor of 22, then AA decreases by a factor of 2\sqrt{2}, so the answer is B.
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Topic: Rigid BodiesConcepts: Kinetic energy, Moments of inertia, Parallel-axis theorem Solution: For a disk rotating around its center, we have For a disk rotating around its edge, we have Recall the moment of inertia of a disk is given by By the parallel axis theorem, a disk rotating around its edge has moment of inertia
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Topic: DynamicsConcepts: Data expression, Newton’s laws Solution: By Newton’s 2nd law, we have In terms of FF, so if we plot (a,F)(a, F) data points, we will obtain a line with slope equal to the mass mm. Choosing two of the data points, so the answer is B. Topic: DynamicsConcepts: Data expression, Forces in mechanics
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Topic: CollisionsConcepts: Conservation of linear momentum Solution: By conservation of linear momentum, we have The x-component of this equation is given by as shown in statement V. The y-component of this equation is given by as shown in statement III. Thus, the answer is B.
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Topic: DynamicsConcepts: Circular motion, Forces in mechanics Solution: The normal force provides the centripetal acceleration so we have The friction force must balance gravity and is bounded by the threshold for static friction, Substituting in the normal force, so the answer is C.
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Topic: CollisionsConcepts: 1D inelastic collision Solution: Letting the rightward direction be positive, the system has initial momentum The system has final momentum Conserving momentum, so the answer is A.
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Topic: KinematicsConcepts: Projectile motion Solution: Recall the range equation from kinematics, where L=v2/gL=v^2/g is the maximum range (when θ=π/4\theta=\pi/4). In our case, so the answer is A.
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Topic: KinematicsConcepts: Motion graphs Solution: On a velocity-time graph, the displacement is given by the area under the curve. Thus, the maximum displacement occurs when the velocity goes to zero (at t=2.5st=2.5\,\mathrm{s}). From the graph, we have Δx=7m\Delta x=7\,\mathrm{m} so the answer is D. Topic: KinematicsConcepts: Motion graphs Solution: Since the acceleration is given by
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Topic: KinematicsConcepts: Motion graphs Solution: The instantaneous velocity on a position-time graph is given by the slope at that point. We have so the answer is A.
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Topic: KinematicsConcepts: Vectors Solution: By the Pythagorean theorem, if a cube has side length ll, the displacement between opposite corners is In our case, l=3ml=3\,\mathrm{m} so d=33md=3\sqrt{3}\,\mathrm{m} and the answer is C.
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Topic: KinematicsConcepts: 5 kinematics equations Solution: Recall our kinematics equation, We have so the answer is D.
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Topic: EnergyConcepts: 1D elastic collision, 1D inelastic collision, Energy dissipation Solution: For the perfectly inelastic collision, we conserve linear momentum to find the velocity of both objects after the collision, Since the combined mass breaks the cord and ends up at rest, all the kinetic energy became spring potential energy (right before it breaks), In
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Topic: DynamicsConcepts: Angular momentum, Circular motion, Work-energy theorem Solution: Note that the velocity is always perpendicular to the tension force by conservation of string, since having a parallel component would mean the total string length is changing (not just the unwound portion). Because perpendicular forces do no work, the object has a constant kinetic energy
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Topic: Rigid BodiesConcepts: Moments of inertia Solution: Recall the moment of inertia of a rod about its center is We are given I=md2I=md^2 which means so the answer is D. Topic: Oscillatory MotionConcepts: Parallel-axis theorem, Physical pendulum Solution: Recall the angular frequency of a physical pendulum is given by where ss is the distance between
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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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