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Conservation of Energy

Energy cannot be created or destroyed in any transformation. This powerful accounting principle helps us analyze everything from particle collisions, to the motion of pendulums.

Mixed Conservative and Non-Conservative Forces - Problem Solving

         

Consider the situation where a baseball player with mass \( m = 76 \text{ kg} \) slides to a stop on level ground. Using energy considerations, calculate the distance \( d \) the baseball player slides, given that his initial speed is \( v = 8 \text{ m/s} \) and the force of friction against him is a constant \( f= 304 \text{ N}. \)

In the above diagram, the bob \( A \) of a pendulum with mass \( m = 8 \text{ kg} \) is released from height \( h = 8 \text{ m}. \) Because of air resistance, the speed of velocity at the lowest position is \( v = 6 \text{ m/s}.\) Find the loss in the total mechanical energy of the bob.

The gravitational acceleration is \( g =10 \text{ m/s}^2. \)

A roller-coaster of mass \( M= 2000 \text{ kg} \) has a speed of \( v_1 = 10 \text{ m/s} \) as it goes over the top of a \(15\)-meter-high hill. Then it goes over a another hill of height \(10\) meters, and at the crest of that hill, the roller-coaster is moving at \( v_2 = 13 \text{ m/s}.\) How much work did friction do on the coaster during its movement between the two hills?

The gravitational acceleration is \( g= 10 \text{ m/s}^2. \)

A skier with mass \( m = 77 \text{ kg} \) coasts up an inclination with angle \( \theta = 45 ^\circ \) and height \( h = 0.5 \text{ m} \) at an initial speed of \( v_i = 12\text{ m/s}. \) Find the square of her final speed at the top, given that the coefficient of kinetic friction between her skis and the snow is \( \mu = 0.5. \)

The gravitational acceleration is \( g = 10 \text{ m/s}^2. \)

An object with mass \( m = 1 \text{ kg} \) at the top of a \( h = 7 \text{ m} \) tall hill starts to go down from rest. The downhill portion of the run is covered with frictionless material but the flat section at the base of the hill has a spot of friction \( 2 \text{ m} \) in length. The object comes straight down the hill, over the spot of friction at the bottom, hits and presses the the damping spring \( 20 \text{ cm} \) in the process. If the spring constant is \( k = 2 \times 10^2 \text{ N/m}, \) what is the magnitude of kinetic friction between the object and flat section.

The gravitational acceleration is \( g = 10 \text{ m/s}^2 \) and air resistance is negligible.

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