College Physics I PHYS 1301.001 - Homework 9 and Exam 2 Notes

College Physics I PHYS 1301.001

  • Homework 9:
    • Due on Mastering Physics, Monday, April 7.
  • Exam 2:
    • In class, April 17.
  • Review/Discussion Sessions (Dr. Khan):
    • SCI 3.265
    • Wed 11 am to 12 pm
    • Thu 1 pm to 2 pm

Supplementary Instruction

  • Tahir Ali:
    • Tuesday at 4-5:15 (MC 3.606B)
    • Thursday at 4-5:15 (MC 3.606B)
  • Physics! Help! Online Problem Session (OPS) on Teams
    • Friday 3 pm
  • Crosby Pineda, Crosby.Pineda@utdallas.edu
    • Monday 1 pm - 2 pm SCI 3.253
    • 2 pm- 3 pm SCI 3.265
  • Muhammad Khalid, Muhammad.Khalid@utdallas.edu
    • Tuesday and Thursday 12 pm - 1 pm, SCI 2.179
  • Meghraj Magadi Shivalingaiah, Meghraj.MagadiShivalingaiah@utdallas.edu
    • Friday 2 pm - 3 pm, SCI 2.112
  • Anusha Srivastava, Anusha.Srivastava@utdallas.edu
    • Monday 2 pm - 3 pm, SCI B.139
  • Mahammed Patel, mxp230007@utdallas.edu
    • Tuesday 10.30 am to 11.30 am and 1.00 pm to pm, SCI 3.253
  • Rittik Patra, rxp200009@utdallas.edu
    • Thursday 2:30 pm-3:30 pm, SCI 2.112

Work and Energy (Chapter 7)

  • Overview of Energy
  • Work
  • Work and Kinetic Energy
  • Work Done with a Varying Force
  • Potential Energy
  • Conservation of Energy
  • Conservative and Non-Conservative Forces
  • Power

Work and Energy with Varying Forces

  • Many forces are not constant.
  • Suppose a particle moves along the x-axis from x1 to x2.
  • The total work done by the force is represented by the area under the curve between the initial and final positions on a graph of force as a function of position.

Stretching a Spring

  • Force due to elongation/compression of a spring follows Hooke’s law:
    • This is a prime example of a varying force.
    • Hooke's Law: F = -kx where:
      • F is the force exerted by the spring.
      • k is the spring constant (a measure of the stiffness of the spring).
      • x is the displacement from the equilibrium position.
  • The work done by stretching/compressing a spring is equal to the area of the shaded triangle:
    • W = \frac{1}{2}kx^2

Work Done on a Spring Scale

  • Problem:
    • A woman weighing 600 N steps onto a bathroom scale containing a heavy spring. The spring compresses by 1.0 cm under her weight.
    • Find:
      • The force constant of the spring.
      • The total work done on it during the compression.

Solution for Spring Scale Problem

  • Choose +x to be upwards (elongation).
  • x = -0.01 m when F_x = -600 N
  • Use Hooke’s Law to find k.
    • F = -kx
    • -600 = -k(0.01)
    • k = 6.0 \times 10^4 N/m
  • W = 3.0 Nm = 3.0 J

Potential Energy

  • Potential Energy: Energy associated with the position of a system rather than its motion
  • Gravitational Potential Energy:
    • When a particle is in the gravitational field of the earth, there is a gravitational potential energy associated with the particle.
    • As the basketball descends, gravitational potential energy is converted to kinetic energy, and the basketball’s speed increases.

Gravitational Potential Energy

  • The quantity, product of the weight mg and the height y above the origin of coordinates, is gravitational potential energy.
    • U_{grav} = mgy
  • We can describe the work done from y1 to y2.
    • W{grav} = mgy1 - mgy_2
  • The negative sign is important:
    • When the body moves up, y increases, W{grav} is negative, and gravitational potential energy increases (\Delta U{grav} > 0).
    • When the body moves down, y decreases, W{grav} is positive, and gravitational potential energy decreases (\Delta U{grav} < 0).

Conservation of Mechanical Energy (gravitational forces only)

  • The total mechanical energy of a system is the sum of its kinetic energy and potential energy.
    • E = K + U
  • A quantity that always has the same value is called a conserved quantity.
  • When only the force of gravity does work on a system, the total mechanical energy of that system is conserved.
    • K1 + U1 = K2 + U2
  • This is an example of the conservation of mechanical energy.

Example 7.7 Conservation of Mechanical Energy (gravitational forces only)

  • You throw a 0.145-kg baseball straight up, giving it an initial velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

Solution for Baseball Example

  • After a baseball leaves your hand, mechanical energy E = K + U_{grav} is conserved.
  • Given:
    • Mass, m = 0.145 kg
    • Initial velocity, v_1 = 20.0 m/s
  • Energy at y_1 (initial position):
    • y_1 = 0
    • E = K1 + U{grav, 1}
  • Energy at y_2 (highest point):
    • v_2 = 0
    • E = K2 + U{grav, 2}
  • Conservation of energy:
    • K1 + U{grav, 1} = K2 + U{grav, 2}
    • \frac{1}{2}mv1^2 + mgy1 = \frac{1}{2}mv2^2 + mgy2
    • Since y1 = 0 and v2 = 0:
      • \frac{1}{2}mv1^2 = mgy2
  • Solving for y_2:
    • y2 = \frac{v1^2}{2g} = \frac{(20.0 \text{ m/s})^2}{2(9.80 \text{ m/s}^2)} = 20.4 \text{ m}

Work and Energy Along a Curved Path

  • We can use the same expression for gravitational potential energy whether the body’s path is curved or straight.
    • W{grav} = mgy1 - mgy_2
  • The total work done by the gravitational force depends only on the difference in height between the two points of the path.
    • Unaffected by any horizontal motion that may occur.
    • mg\Delta s \cos{\beta} = -mg\Delta y
  • We can use the same expression for gravitational potential energy whether the body’s path is curved or straight.