Practice · Unit 7
Gravitation and periodic motion exercises
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The page offers a short, varied set. There is no overall goal to complete.
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Universal gravitation
Exercise to explore
Force between two masses
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
Masses of 1000 kg and 1500 kg are 2.0 m apart center to center. Use G=6.674×10^-11 N·m²/kg². Calculate |F|.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Universal gravitation
Exercise to explore
Distance ratio
- Type
- Conceptual
- Difficulty
- 1/5
- Time
- 5 min
If only separation changes and r doubles, gravitational force becomes:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Universal gravitation
Exercise to explore
Superposition on a line
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
A test mass m=2.0 kg is at x=0. M₁=5.0×10^10 kg is at x=-2.0 m and M₂=2.0×10^10 kg at x=+1.0 m. Use G=6.674×10^-11 and +x to the right. Calculate net F_x.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Universal gravitation
Exercise to explore
Gravitational third law
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
A planet attracts a moon. How do the magnitudes of their mutual gravitational forces compare?
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Field and weight
Exercise to explore
Field at two radii
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
At a planet's surface g₀=9.8 m/s². Calculate g at r=2R.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Field and weight
Exercise to explore
Weight at 1.5 radii
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
A 70 kg person is at r=1.5R_E. Use g₀=9.8 m/s². Calculate gravitational weight.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Field and weight
Exercise to explore
Mass and weight
- Type
- Conceptual
- Difficulty
- 1/5
- Time
- 5 min
A person travels from Earth to a body with lower g. What happens ideally?
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Field and weight
Exercise to explore
Field at the midpoint
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
Two identical fixed masses have a test mass exactly midway between them. Ignoring other masses, the net gravitational field there is:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Gravitational energy
Exercise to explore
Earth gravitational potential energy
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
A 1000 kg mass is at r=7.00×10^6 m from Earth's center. Use μ_E=3.986×10^14 m³/s² and U(∞)=0. Calculate U.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Gravitational energy
Exercise to explore
Radial potential-energy change
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
A 1000 kg mass moves from R_E to 2R_E. Use μ_E=3.986×10^14 m³/s² and R_E=6.371×10^6 m. Calculate ΔU.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Gravitational energy
Exercise to explore
Escape-energy limit
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
With U(∞)=0, an object launched at exactly the ideal escape speed reaches infinity with zero final speed. Its total mechanical energy is:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Gravitational energy
Exercise to explore
Local mgh approximation
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
When is ΔU≈mgΔh reasonable near a planet?
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Orbits and satellites
Exercise to explore
Circular-orbit speed
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
An Earth satellite is in a circular orbit of radius r=7.00×10^6 m. Use μ_E=3.986×10^14 m³/s². Calculate v_orb.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Orbits and satellites
Exercise to explore
Circular-orbit period
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
For a circular Earth orbit with r=7.00×10^6 m and μ_E=3.986×10^14 m³/s², calculate T.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Orbits and satellites
Exercise to explore
Specific orbital energy
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
For a circular Earth orbit at r=7.00×10^6 m, use μ_E=3.986×10^14 m³/s². Calculate E/m.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Orbits and satellites
Exercise to explore
Escape speed
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
At r=7.00×10^6 m around Earth, use μ_E=3.986×10^14 m³/s². Calculate v_esc.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Kepler and limits
Exercise to explore
Kepler scaling
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
Two bodies orbit the same central mass. If a₂=4a₁, calculate T₂/T₁.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Kepler and limits
Exercise to explore
Speed in an elliptical orbit
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
A planet in an elliptical orbit moves:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Kepler and limits
Exercise to explore
Period from Kepler's third law
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
A satellite has semimajor axis a=4.00×10^8 m around Earth. Use μ_E=3.986×10^14 m³/s². Calculate T.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Kepler and limits
Exercise to explore
Sun's Schwarzschild radius
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
As relativistic context, calculate for G=6.674×10^-11, M=1.989×10^30 kg, and c=3.00×10^8 m/s.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillations
Exercise to explore
Period from frequency
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
An oscillation has f=2.50 Hz. Calculate T.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillations
Exercise to explore
Angular frequency
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
An oscillation has T=0.800 s. Calculate ω.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillations
Exercise to explore
Amplitude from extrema
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
A particle oscillates from x=-0.12 m to x=+0.12 m about x=0. Calculate A.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillations
Exercise to explore
Periodic does not imply SHM
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
Which statement is correct?
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Simple harmonic motion
Exercise to explore
Natural spring frequency
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
m=2.0 kg and k=200 N/m. Calculate ω₀.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Simple harmonic motion
Exercise to explore
Position at a quarter cycle
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
x(t)=0.10 cos(10t) m. Calculate x at t=π/20 s.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Simple harmonic motion
Exercise to explore
Maximum SHM speed
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
An SHM has A=0.10 m and ω₀=10 rad/s. Calculate v_max.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Simple harmonic motion
Exercise to explore
Acceleration at a position
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
An SHM has ω₀=10 rad/s. At one instant x=+0.050 m. Calculate a.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillator energy
Exercise to explore
Total oscillator energy
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
k=100 N/m and A=0.20 m. Calculate total mechanical energy.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillator energy
Exercise to explore
Elastic potential energy
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
k=100 N/m and x=0.10 m. Calculate U with U=0 at equilibrium.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillator energy
Exercise to explore
Speed from energy
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
m=1.0 kg, k=100 N/m, A=0.20 m, and x=0.10 m. Calculate |v|.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Oscillator energy
Exercise to explore
Speed and position
- Type
- Conceptual
- Difficulty
- 1/5
- Time
- 5 min
Where is speed greatest in an ideal mass–spring SHM?
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Pendulums
Exercise to explore
Simple-pendulum period
- Type
- Numerical
- Difficulty
- 1/5
- Time
- 5 min
A simple pendulum with L=1.00 m oscillates through a small angle. Use g=9.8 m/s². Calculate T.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Pendulums
Exercise to explore
Pendulum length
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
What length gives a small-angle simple pendulum T=2.00 s? Use g=9.8 m/s².
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Pendulums
Exercise to explore
Rod as a physical pendulum
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
A uniform rod of L=1.00 m oscillates about one end with small amplitude. Use I_p=(1/3)ML², d=L/2, and g=9.8 m/s². Calculate T.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Pendulums
Exercise to explore
Amplitude limit
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
The statement “a simple pendulum's period does not depend on amplitude” is:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Damping and resonance
Exercise to explore
Damped envelope
- Type
- Numerical
- Difficulty
- 2/5
- Time
- 5 min
A₀=0.20 m, m=1.0 kg, and b=2.0 kg/s. Calculate A_env=A₀e^{-bt/(2m)} at t=1.0 s.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Damping and resonance
Exercise to explore
Damped frequency
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
m=1.0 kg, k=100 N/m, and b=4.0 kg/s. Calculate ω_d=√[k/m-(b/2m)²].
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Damping and resonance
Exercise to explore
Steady-state amplitude
- Type
- Numerical
- Difficulty
- 3/5
- Time
- 5 min
m=1.0 kg, k=100 N/m, b=4.0 kg/s, F₀=10 N, and Ω=10 rad/s. Calculate the steady-state amplitude.
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
- Interpretation
Check the result against the model conditions and a limiting case.
Damping and resonance
Exercise to explore
Damping and resonance
- Type
- Conceptual
- Difficulty
- 2/5
- Time
- 5 min
As damping increases for a driven linear oscillator, the amplitude peak normally:
Request a hint
- State the reference, direction, and model conditions before substituting.
Review the solution
- Principle
Select the governing relationship and state its validity conditions.
- Calculation
Substitute the supplied data while preserving signs and units.
Universal gravitation
Field and weight
Gravitational energy
Orbits and satellites
Kepler and limits
Oscillations
Simple harmonic motion
Simple harmonic motion
Oscillator energy
Pendulums
Damping and resonance
Damping and resonance