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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.

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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 \( g =10 \text{ m/s}^2. \)

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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 \( g= 10 \text{ m/s}^2. \)

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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 \( g = 10 \text{ m/s}^2. \)

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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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