Energy and Conservation of Energy Lab Study Guide

Kinetic Energy ( $K$ ): For a rolling ball on a track, the kinetic energy is expressed by the formula: $K = \frac{7}{10} m v^2$
Potential Energy ( $U$ ): The gravitational potential energy of the ball is defined as: $PE = mgh$

Kinetic Energy Maxima and Minima: * Largest Kinetic Energy: This occurs at the lowest point of the track. At this position, the ball has reached its maximum speed while rolling down the incline. * Smallest Kinetic Energy: This occurs at the highest point of the path. At this peak, the ball is at its slowest or momentarily stationary, having not yet gained speed from the descent.
Potential Energy Maxima and Minima: * Largest Potential Energy: Located at the absolute top of the system (e.g., right before a roller coaster drops). Height is at its maximum here. * Smallest Potential Energy: Located at the point where the speed is greatest, typically the lowest vertical elevation in the system.

Force and Height Relationships: * Total Energy Trends: Observations indicate that total energy values decrease by approximately one unit for every two height increments. * Speed vs. Height: When the ball is at its maximum height, it traverses the ramp more slowly. Conversely, when the height is lower, the speed through that section is faster. * Speed Localization: The speed of the ball is greatest at the bottom or just before entering a loop. It is at its minimum at the highest point of the ramp or at the very beginning of the descent. * Impact of Direction: Height is identified as the primary factor affecting speed. While a marble moving downhill at a constant speed remains stable, entering an uphill section (like a loop) results in a decrease in speed.

Symmetry of Motion: In a frictionless environment (like the U-shaped track), the maximum height reached on the left side of the track is identical to the maximum height reached on the right side.
Energy Bar Chart Dynamics: * Kinetic Energy ( $KE$ ): Reaches its maximum value at the exact bottom of the loop. * Potential Energy ( $PE$ ): Reaches its maximum at the highest points on either side of the track. * Speedometer Readings: At the uppermost positions (the turning points), the speed is exactly $0\,m/s$ . Maximum speed is achieved at the exact bottom of the track.
Relationship Between Speed and Kinetic Energy: * Speed and kinetic energy are directly proportional. When speed is at its maximum, kinetic energy is at its maximum. * If speed is $0\,m/s$ , kinetic energy is likewise zero. * Any increase or decrease in speed results in a corresponding increase or decrease in kinetic energy.
Motion Diagrams (Path Tracking): * The dots (representing equal time intervals) are furthest apart at the very bottom of the path. * Theoretical Explanation: This follows the formula $\text{speed} = \frac{\text{distance}}{\text{time}}$ . Because the time between dots is constant, a greater distance between dots indicates the skater traveled further in that interval, thus possessing higher speed.

Height: Higher elevation leads to increased potential energy. Lowering the elevation decreases potential energy.
Gravity: Potential energy is directly related to the gravitational constant. Increasing the gravity setting on the simulation increases potential energy, whereas decreasing gravity reduces it.
Mass: There is a direct relationship between mass and energy. As the mass of the skater increases, both potential energy and kinetic energy increase.
Reference Height: The reference height line (dotted line) determines the zero-point for potential energy. If the reference line is raised above the skater's position, the bar chart will display negative potential energy.

Energy Transformation: When friction is enabled, kinetic and potential energy are transformed into thermal energy.
Irreversibility: Thermal energy dissipates into the environment. It cannot naturally transform back into kinetic or potential energy.
System Equilibrium: With friction active, the skater will eventually lose all mechanical energy and stop moving as the energy is converted to heat.
Alternative Source of Thermal Energy: Thermal energy can also be increased by dropping the skater from a significant height directly onto the track rather than letting them slide from a starting point.

Angle ( $^{\circ}$ )	Force ( $N$ )	Displacement ( $m$ )	Work ( $J$ )
$30^{\circ}$	$9.8\,N$	$2\,m$	$19.6\,J$
$40^{\circ}$	$12.6\,N$	$1.56\,m$	$19.6\,J$
$50^{\circ}$	$15\,N$	$1.31\,m$	$19.6\,J$
$60^{\circ}$	$17\,N$	$1.15\,m$	$19.6\,J$
$70^{\circ}$	$18.4\,N$	$1.06\,m$	$19.6\,J$
$80^{\circ}$	$19.3\,N$	$1.02\,m$	$19.6\,J$
$90^{\circ}$	$19.6\,N$	$1\,m$	$19.6\,J$

Effect of Incline Angle: * On Force: As the incline angle increases ( $30^{\circ} \rightarrow 90^{\circ}$ ), the force required to pull the cart up the hill increases (from $9.8\,N$ to $19.6\,N$ ). * On Work: The incline angle does not affect the total work done. The work remains constant at $19.6\,J$ for all angles because the increase in force is balanced by a decrease in displacement.
Work-Energy Connection: * The work performed by the applied force causes a change in potential energy. As the cart rises to a higher elevation, its gravitational potential energy increases. * Potential Energy Calculation: Using a mass of $2.0\,kg$ , gravity ( $g$ ) of $9.8\,m/s^2$ , and a vertical height of $1.0\,m$ : $PE = 2.0 \times 9.8 \times 1.0 = 19.6\,J$ * Conservation of Energy: The calculated potential energy ( $19.6\,J$ ) is equal to the work recorded in the data table ( $19.6\,J$ ), demonstrating the principle of conservation of energy.

Comparison of Work and Potential Energy: * Question: How does the calculated potential energy compare to the work values in the data table? * Response: They are equal ( $19.6\,J$ ). This equivalence proves that the work put into the system is stored as potential energy.
Speed and Kinetic Energy Correlation: * Question: In what way is Speed related to Kinetic Energy? * Response: They move in tandem. Kinetic energy increases when speed increases and decreases when speed decreases. If speed is zero, kinetic energy must also be zero.
Dots in Motion Diagrams: * Question: When are the dots furthest apart and what does this indicate? * Response: They are furthest apart at the bottom of the track. This indicates that the skater is moving at their fastest speed at that point because they cover more distance in the same time interval.
Thermal Energy Dissipation: * Question: If you wait long enough, will Thermal Energy transform back into Kinetic and Potential Energy? * Response: No. Thermal energy represents energy that has effectively left the mechanical system and cannot be reclaimed to produce motion or height within this context.