Lecture Notes on Kinetic Energy and Work
Today's Learning Goals
Understand the concept of Work.
Learn about the Work-Kinetic Energy Theorem.
Recap on Kinetic Energy
Equivalent Units of Kinetic Energy:
Newtons * m
Joules
kg * m² * s⁻²
All of the above.
Work
Definition:
Work is the energy transferred to or from an object by a force acting on it.
Positive vs Negative Work:
Positive work: Energy is transferred to the object.
Negative work: Energy is transferred from the object.
Mathematical Expression: W = rac{dW}{dr} F \cdot dr
Where F is the force vector and dr is the displacement vector.
Dot Product
Definition: The dot product (or scalar product) results in a scalar from two vectors.
Mathematical Representation:
a \cdot b = |a||b| \cos(\theta)Calculating Dot Product: a \cdot b = axbx + ayby + azbz
The sign of a \cdot b depends on the angle \theta between the vectors.
Work by a Constant Force
When force is constant, work can be calculated as: W = F \cdot \Delta r
For a constant force F :
W = Fx \Delta x + Fy \Delta y + F_z \Delta z
Gravitational Force and Work
Work done by gravitational force can be determined by the direction of motion:
Moving Up the Slope: Work done by gravitational force is negative.
Moving Down the Slope: Work done by gravitational force is positive.
Mathematical expression for work done by gravity when moving down an incline of angle \alpha :
W = -mg|\Delta r| \sin(\alpha)
Work-Kinetic Energy Theorem
Statement: The net work done by all forces on an object equals the change in its kinetic energy.
Expressed as: W{NET} = Kf - K_i
Where Kf is final kinetic energy and Ki is initial kinetic energy.
The total work done can also be calculated as:
W{NET} = \sum Wi = W1 + W2 + … + W_n
Example Problems
Baseball Example:
A pitcher throws a 0.5 kg baseball from rest to an exit speed of 50 m/s. The net work done on the baseball can be calculated using the Work-Kinetic Energy theorem.
Gravity Work Calculation:
For a climber of mass m climbing a wall, work done by gravity as they climb from r1 to r2 is given by:
W = -mgΔy
Displacement Example:
For a climber with $65$ kg mass moving a displacement of $2m$ horizontally and $5m$ vertically, compute the work done against gravity.
Kinetic Energy Comparison:
Determine which of three slides results in the highest kinetic energy at the bottom based on conservation of energy principles (potential energy converted to kinetic energy).
Conclusions
Work and kinetic energy are interrelated concepts in mechanics.
Understanding the mathematical expressions and real-world implications of these concepts is critical for problem-solving in physics.
The sign of work done (positive, negative, or zero) is determined by the direction of force relative to the displacement.
Key Equations and Concepts
Work
Definition: Work is the energy transferred to or from an object by a force acting on it.
Mathematical Expression: W = \frac{dW}{dr} F \cdot dr
Where F is the force vector and dr is the displacement vector.
Dot Product
Definition: The dot product results in a scalar from two vectors.
Mathematical Representation: a \cdot b = |a||b| \cos(\theta)
Calculating Dot Product: a \cdot b = axbx + ayby + azbz
Work by a Constant Force
When force is constant, work can be calculated as: W = F \cdot \Delta r
For a constant force F : W = Fx \Delta x + Fy \Delta y + F_z \Delta z
Gravitational Force and Work
Direction Matters:
Moving Up the Slope: Work done by gravitational force is negative.
Moving Down the Slope: Work done by gravitational force is positive.
Expression for Work Done by Gravity (Moving Down an Incline of Angle \alpha ): W = -mg|\Delta r| \sin(\alpha)
Work-Kinetic Energy Theorem
Statement: The net work done by all forces on an object equals the change in its kinetic energy.
Expressed as: W{NET} = Kf - Ki
Total Work Done: W{NET} = \sum Wi = W1 + W2 + … + W_n
Conclusions
Work and kinetic energy are interrelated concepts in mechanics.
Understanding the mathematical expressions and real-world implications of these concepts is critical for problem-solving in physics.