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PY 211: Introductory Physics I – Spring 2023

Homework 5

Due 11:59 PM Friday February 24, 2023

Potential Energy

MasteringPhysics Online Homework:

(1-5) [5 points total] A selection offive short conceptual/reading questions, 1 point each.

(6) [5 points] Bucket, box, and bag.

(7) [5 points] Skiing into a snow drift

(8) [5 points] Fun with a spring gun

Written Homework:

(Hand in via Gradescope. Your work will be graded on the quality of the solution presented, not just the right answer. Neatly show your work, use diagrams where appropriate. One page per question is recommended. Posting to Chegg or other services is a violation of Boston University Academic Conduct and U.S. copyright law.

(9) [10 points] Consider a 1-dimensional potential given by the following. Because U(x) → 0 as x → ∞, it makes sense to define U(∞) = 0.

U(x) = − , − + 0

(a) What is the force equation associated with this potential energy function? Provide a symbolic expression.

(b) Graph the potential energy function. A computer software generated graph is preferable, a very neatly hand drawn graph on graph paper is acceptable. Take a = 1.0 m and A = 64.0 J. Note that the potential energy blows up at x = 0 so you must restrict the range to see the function nicely. Try graphing the function from x = 0 to 16 m and U(x) from 0 to -4 J, or thereabouts.

Label your axes.

(c) Find locations of equilibrium and identify if they are stable or unstable.

(d) Is this a conservative force? How do you know?

(e) Place a particle of mass m = 0.2 kg at rest at the unstable equilibrium point closest to the origin. Displace slightly it in the direction of nearest stable equilibrium point. When it reaches the stable equilibrium point, what is the speed of the particle?

(10) [10 points] You use a spring to impart initial speed to a small ball that you will launch upward. The spring does not obey Hooke’s Law perfectly, it follows:

F(X) =    − kX + bX2

Recall that Hooke’s Law provides the force of the spring reacting to being compressed ( −X) or

(a) To compress the spring 5.0 cm, how much total external force is required?  And to stretch it by 5.0 cm, how much external force? Which direction requires larger magnitude of force?

(b) Find a symbolic expression for the potential energy function. Take U#  = 0 at the relaxed length of X = 0.

(c) The spring is compressed 5 cm and is used to launch a 0.02 kg ball, find the energy stored in the spring and the launch speed.

(d) In flight, the ball loses -0.04 joules of energy to drag (air resistance). How high does the ball go? Use g = 9.81 m/s2 .