Part A = 1/2k((-L – x)^2 + (L – x)^2 + 2(-y)^2)

Part B = 2 * kxx(hat) + -2 * kyy(hat)

What is the potential energy of the two-spring system after the point of connection has been moved to position (x, y)?

Use the potential energy expression from Part A to find the force F on the junction point, the point where the two springs are attached to each other.

Part A = F = mg/2

Part B = W_{d} / W_{p} = 1.00

What is the magnitude F of the upward force you must apply to the rope to start raising the box with constant velocity?

What is W_{d} / W_{p}, the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?

Part A = smooth

Part B = a distance 2h/3 above the floor

Part C = -2mgh/3

Part D = -mgh

Part E = 1/2mv_{i}^2 + mgh_{i} = 1/2mv_{f}^2 + mgh_{f}

Part F = K increases; U decreases; E stays the same

Part G = sqrt(v^2 + 2gh)

Part H = 1/2mv_{i}^2 + W_{nc} = 1/2mv_{f}^2

Part I = K decreases; U stays the same; E decreases

Part J = friction

Part K = 0.5mv^2 + mgh

Part A = 455.165 km

Part B = 8.9523 g

Part C = 3.1107 g

Consider a bird that flies at an average speed of 10.7 m/s and releases energy from its body fat reserves at an average rate of 3.70 W (this rate represents the power consumption of the bird). Assume that the bird consumes 4g of fat to fly over a distance d_{b} without stopping for feeding. How far will the bird fly before feeding again?

How many grams of carbohydrate m_{carb} would the bird have to consume to travel the same distance d_{b}?

Field observations suggest that a migrating ruby-throated hummingbird can fly across the Gulf of Mexico on a nonstop flight traveling a distance of about 800 km. Assuming that the bird has an average speed of 40.0 km/hr and an average power consumption of 1.70 W, how many grams of fat does a ruby-throated hummingbird need to accomplish the nonstop flight across the Gulf of Mexico?

Part A = L = ((0.5 * x_{c}^2 * k) – m * g * sin(θ) * x_{c}) / (m * g * (sin(θ) + cos(θ)*μ))

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A block of mass m is placed in a smooth-bored spring gun at the bottom of the incline so that it compresses the spring by an amount x_{c}. The spring has spring constant k. The incline makes an angle θ with the horizontal and the coefficient of kinetic friction between the block and the incline is μ. The block is released, exits the muzzle of the gun, and slides up an incline a total distance L.

Find L, the distance traveled along the incline by the block after it exits the gun. Ignore friction when the block is inside the gun. Also, assume that the uncompressed spring is just at the top of the gun (i.e., the block moves a distance x_{c} while inside of the gun). Use g for the magnitude of acceleration due to gravity.

Part A = x_{2} = d / sqrt(2kH / mg)

Part B = x_{2} = (x_{1} * d) / (d – d_{12})

Two children are trying to shoot a marble of mass m into a small box using a spring-loaded gun that is fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height H above the top of the box (the height of which is negligibly small), and the center of the box is a distance d from the edge of the table. The spring has a spring constant k. The first child compresses the spring a distance x_{1} and finds that the marble falls short of its target by a horizontal distance d_{12}.

By what distance, x_{2}, should the second child compress the spring so that the marble lands in the middle of the box? (Assume that height of the box is negligible, so that there is no chance that the marble will hit the side of the box before it lands in the bottom.)

Now imagine that the second child does not know the mass of the marble, the height of the table above the floor, or the spring constant. Find an expression for x_{2} that depends only on X_{1} and distance measurements.