Physics 2201 Homework III Part 1

Phy 2201 Page 1 Physics 2201 Homework Iii Part 1

Analyze the following physics problems related to work, energy, and forces acting on objects. Show all work with clear setup and explanations, and base solutions on work and energy methods.

Paper For Above instruction

The assignment includes multiple problems covering topics such as work done by friction and gravity, energy conversion due to friction, force analyses on sliding beads, work done by gravity along curved paths, and energy considerations in pendulum motion. Each problem requires careful diagramming, mathematical derivation, and conceptual reasoning based on the principles of work, energy conservation, and the dynamics of particles on curved paths.

Problem 1: Work done by friction and gravity on a sliding block

A wooden block slides off a level table with an initial speed of 2.3 m/s. The coefficient of kinetic friction between the table and the block is 0.22. The questions revolve around calculating the work done by friction and gravity between the initial point (A) and the point just before hitting the ground (B), followed by using the work-energy theorem to find the kinetic energy at B.

First, the work done by friction can be calculated as:

W_friction = - μ_k m g * d

where μ_k is the coefficient of kinetic friction, m is the mass of the block, g is acceleration due to gravity, and d is the length of the slide. The negative sign indicates that friction opposes motion.

The work done by gravity is determined by considering the change in gravitational potential energy plus any work done as the mass moves horizontally and vertically.

Applying the work-energy theorem, the change in the kinetic energy of the block from point A to B equals the net work done by all forces, i.e.,

ΔKE = W_gravity + W_friction

This allows calculating the kinetic energy at B, considering initial KE at A, work done by gravity (which may be positive or negative depending on the motion), and work done by friction.

Problem 2: Energy and force analysis on a bead sliding on a wire

A bead of mass 0.06 kg slides on a bent wire where point A is 0.56 m higher than point C. Friction acts on the bead as it slides from A to C, and the initial and final velocities are given at these points. The problem involves drawing force diagrams, determining work by forces, and calculating energy conversions.

Part a involves drawing free body diagrams at points A, B, and C, identifying the forces such as gravity, normal force, and friction. Forces doing work are typically the component of gravity along the path unless friction acts tangentially, converting mechanical energy to thermal energy.

Part b asks for the energy lost to non-mechanical forms, which equals the work done by friction — the energy difference arising due to frictional dissipation.

Problem 3: Work done by gravity along a parabolic wire

A bead of mass m slides on a wire described by y = c - bx + ax2. The displacement vector dr along the wire depends on dx and the derivatives of y with respect to x. The problem involves calculating the work done by gravity along this curved path.

Part a requires calculating the dot product F_g · dr, where F_g = -mg ĵ, with dr expressed in terms of dx and derivatives of y, which yields an expression for the infinitesimal work done by gravity over the wire segment.

Part b involves integrating this dot product from x1 to x2 to find the total work done during the bead's sliding motion.

Part c demonstrates the algebraic equivalence of this work with the change in gravitational potential energy, highlighting that the work done by gravity depends only on the vertical displacement Δy, confirming the principle that gravitational work depends on change in height.

Problem 4: Energy considerations in a pendulum with external force

A pendulum with a sphere of mass m is raised through a vertical distance of 0.8 m by an external force Fext that varies in magnitude and direction while moving at constant speed along a circular arc.

Part a involves identifying all forces: gravity, tension, and external force, considering their directions and whether the external force balances gravity to maintain constant speed. The question challenges the misconception that the external force is always equal and opposite to gravity, clarifying the nature of work done.

Part b investigates which forces do work (notably, the external force may do work, while tension generally does not in uniform circular motion), and whether the kinetic energy of the sphere changes (it does not if speed is constant).

Part c relates the work done by gravity and the external force, emphasizing the energy transfer, where gravity gains potential energy equal to mgh, and the external force supplies or absorbs energy as needed, with energy conservation including internal and external work contributions.

Part d discusses the increase in gravitational potential energy, the minimum chemical energy expended within the person’s body to perform this work, and how conservation of energy ensures the total energy balance in the system, considering all forms of energy and energy input.

References

  • Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers. W. H. Freeman.
  • Serway, R. A., & Jewett, J. W. (2013). Physics for Scientists and Engineers. Brooks Cole.
  • Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics. Wiley.
  • Knight, R. D. (2013). Physics for Scientists and Engineers: A Strategic Approach. Pearson.
  • Young, H. D., & Freedman, R. A. (2017). University Physics with Modern Physics. Pearson.
  • Feynman, R. P., Leighton, R. B., & Sands, M. (2010). The Feynman Lectures on Physics Vol. 1-3. California Institute of Technology.
  • Reif, F. (2008).fundamentals of Statistical and Thermal Physics. Waveland Press.
  • Kittel, C., Kroemer, H., & Seeger, K. (1980). Thermal Physics. W. H. Freeman.
  • McDonald, J. (2014). Physics: Principles with Applications. Pearson.
  • Gordon, K. (2011). Introductory Physics. University of Toronto Press.

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