AP Free Response Question
1999 C2
Printer Friendly Version
A spherical, nonrotating planet has a radius
R
and a uniform density
throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass
m
could move freely under the influence of gravity. Let
r
be the distance of the ball from the center of the planet.
(a) Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = —Cr, where
(b) On the axes below, sketch the force
F
on the ball as a function of distance
r
from the center of the planet.
The ball is dropped into the tunnel from rest at point P at the planet’s surface.
(c) Determine the work done by gravity as the ball moves from the surface to the center of the planet.
(d) Determine the speed of the ball when it reaches the center of the planet.
(e) Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.
(f) Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
friction
gravitational force (vector)
gravitational potential energy
Hooke's Law
impulse
kinetic energy
linear momentum
linear velocity and angular velocity
moment of inertia
net torque
Newton's 2nd Law
Newton's Law of Universal Gravitation
period and frequency
period of a simple pendulum
period of a spring
potential elastic energy
potential energy
power (dot product)
rate of change of momentum
rate of change of work
rotational kinetic energy
torque
uniform acceleration  displacement and instantaneous velocity
uniform acceleration  instantaneous position
uniform acceleration  instantaneous velocity
work (dot product)
Related Documents
Lab:
Labs 
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Resource Lesson:
RL 
Average Velocity  A Calculus Approach
RL 
Derivatives: Instantaneous vs Average Velocities
RL 
Thin Rods: Center of Mass
Review:
WS 
Drill: Mechanics
WS 
Drill: Waves and Sound
TB 
Schaum's 11th Edition
REV 
Course Objectives
REV 
Cumulative Review
REV 
Drill: Circular Motion
REV 
Drill: Common Variables
REV 
Drill: DC Circuits
REV 
Drill: Dynamics
REV 
Drill: Electrostatics
REV 
Drill: Kinematics
REV 
Drill: Magnetism
REV 
Drill: Metric System
REV 
Drill: Modern
REV 
Drill: Physical Optics
REV 
Drill: Projectiles
REV 
Drill: Refraction and Lenses
REV 
Drill: Rotary Motion
REV 
Drill: SHM
REV 
Drill: Thermodynamics
REV 
Drill: Work and Energy
REV 
Preregistration Survey
REV 
Sample NY Regents Review Questions
Worksheet:
AAPT 
1994 Physics Olympiad Screening Test (Part 1)
AAPT 
1994 Physics Olympiad Screening Test (Part 2)
AAPT 
1994 Physics Quiz Bowl (120)
AAPT 
1994 Physics Quiz Bowl (2140)
AAPT 
1995 Physics Olympiad Screening Test (Part 1)
AAPT 
1995 Physics Olympiad Screening Test (Part 2)
AAPT 
1995 Physics Quiz Bowl (120)
AAPT 
1995 Physics Quiz Bowl (Part 2)
AAPT 
1996 Physics Olympiad Screening Test (Part 1)
AAPT 
1996 Physics Olympiad Screening Test (Part 2)
AAPT 
1996 Physics Quiz Bowl (Part 1)
AAPT 
1996 Physics Quiz Bowl (Part 2)
AAPT 
1997 Physics Olympiad Screening Test (Part 1)
AAPT 
1997 Physics Olympiad Screening Test (Part 2)
AAPT 
1997 Physics Quiz Bowl (Part 1)
AAPT 
1997 Physics Quiz Bowl (Part 2)
AAPT 
1998 Physics Olympiad Screening Test (Part 1)
AAPT 
1998 Physics Olympiad Screening Test (Part 2)
AAPT 
1998 Physics Quiz Bowl (Part 1)
AAPT 
1998 Physics Quiz Bowl (Part 2)
AAPT 
1999 Physics Olympiad Screening Test (Part 1)
AAPT 
1999 Physics Olympiad Screening Test (Part 2)
AAPT 
1999 Physics Quiz Bowl (Part 1)
AAPT 
1999 Physics Quiz Bowl (Part 2)
AAPT 
2000 Physics Olympiad Screening Test (Part 2)
AAPT 
2000 Physics Olympiad Screening Test (Part 2)
AAPT 
2000 Physics Quiz Bowl (2140)
AAPT 
2000 Physics Quiz Bowl (Part 1)
AAPT 
2006 Physics Quiz Bowl (Part 1)
AAPT 
2006 Physics Quiz Bowl (Part 2)
AAPT 
2007 Physics Quiz Bowl (Part 1)
AAPT 
2007 Physics Quiz Bowl (Part 2)
AAPT 
2008 Physics Quiz Bowl (Part 2)
AAPT 
2008 PhysicsBowl (Part 1)
AAPT 
2015 net F = ma Contest
AAPT 
PhysicsBowl 2009 (Part 1)
AAPT 
PhysicsBowl 2009 (Part 2)
AAPT 
PhysicsBowl 2010 (Part 1)
AAPT 
PhysicsBowl 2010 (Part 2)
AAPT 
PhysicsBowl 2011 (Part 1)
AAPT 
PhysicsBowl 2011 (Part 2)
AAPT 
PhysicsBowl 2012 (Part 1)
AAPT 
PhysicsBowl 2012 (Part 2)
AAPT 
PhysicsBowl 2013 (Part 1)
AAPT 
PhysicsBowl 2013 (Part 2)
AAPT 
PhysicsBowl 2014 (Part 1)
AAPT 
PhysicsBowl 2014 (Part 2)
AAPT 
PhysicsBowl 2015 (Part 1)
AAPT 
PhysicsBowl 2015 (Part 2)
AAPT 
PhysicsBowl 2016 (Part 1)
AAPT 
PhysicsBowl 2016 (Part 2)
AAPT 
PhysicsBowl 2017 (Part 1)
AAPT 
PhysicsBowl 2017 (Part 2)
AAPT 
PhysicsBowl 2018 (Part 1)
AAPT 
PhysicsBowl 2018 (Part 2)
AAPT 
PhysicsBowl 2019 (Part 1)
AAPT 
PhysicsBowl 2019 (Part 2)
NY 
January 2006, Part 1
NY 
January 2006, Part 2
NY 
January 2006, Part 3
NY 
January 2007, Part 1
NY 
January 2007, Part 2
NY 
January 2007, Part 3
NY 
January 2008, Part 1
NY 
January 2008, Part 2
NY 
January 2008, Part 3
NY 
January 2008, Part 4
NY 
January 2009, Part 1
NY 
January 2009, Part 2
NY 
June 2006, Part 1
NY 
June 2006, Part 2
NY 
June 2006, Part 3
NY 
June 2007, Part 1
NY 
June 2007, Part 2
NY 
June 2007, Part 3
NY 
June 2008, Part 1
NY 
June 2008, Part 2
NY 
June 2008, Part 3
NY 
June 2008, Part 4
NY 
June 2009, Part 1
NY 
June 2009, Part 2
NY 
June 2010, Part 1
NY 
June 2010, Part 2
NY 
June 2010, Part 3
NY 
June 2011, Part 1
NY 
June 2011, Part 2
NY 
June 2011, Part 3
NY 
June 2012, Part 1
NY 
June 2012, Part 2
NY 
June 2012, Part 3
NY 
June 2013, Part 1
NY 
June 2013, Part 2
NY 
June 2013, Part 3
NY 
June 2014, Part 1
NY 
June 2014, Part 2
NY 
June 2014, Part 3
NY 
June 2015, Part 1
NY 
June 2015, Part 2
NY 
June 2015, Part 3
NY 
June 2016, Part 1
NY 
June 2016, Part 2
NY 
June 2016, Part 3
NY 
June 2017, Part 1
NY 
June 2017, Part 2
NY 
June 2017, Part 3
NY 
June 2018, Part 1
NY 
June 2018, Part 2
NY 
June 2018, Part 3
NY 
June 2019, Part 1
NY 
June 2019, Part 2
NY 
June 2019, Part 3
TB 
Antiderivatives and Kinematics Functions
TB 
Projectile Summary
TB 
Projectile Summary

MCAS 2004 Session 1

MCAS 2004 Session 2

MCAS 2005 Session 1

MCAS 2005 Session 2

MCAS 2006 Session 1

MCAS 2006 Session 2

MCAS 2007 Session 1

MCAS 2007 Session 2

MCAS 2008 Session 1

MCAS 2008 Session 2

MCAS 2009 Session 1

MCAS 2009 Session 2

MCAS 2010 Session 1

MCAS 2010 Session 2

MCAS 2011 Session 1

MCAS 2011 Session 2

MCAS 2012 Session 1

MCAS 2012 Session 2

MCAS 2013 Session 1

MCAS 2013 Session 2

MCAS 2014 Session 1

MCAS 2014 Session 2

MCAS 2015 Session 1

MCAS 2015 Session 2

MCAS 2016 Session 1

MCAS 2016 Session 2

MCAS 2017 Session 1

MCAS 2017 Session 2

MCAS 2018 Session 1

MCAS 2018 Session 2

MCAS 2019 Session 1

MCAS 2019 Session 2
CBETS
Copyright © 19702023
All rights reserved.
Used with
permission
Mainland High School
Daytona Beach, FL 32114