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The 100 kg box shown below is being pulled along the x-axis by a student. The box slides across a rough surface, and its position \(x\) varies with time \(t\) according to the equation \(x = 0.5t^3 + 2t\) , where \(x\) is in meters and \(t\) is in seconds.
A. Determine the speed of the box at time \(t\) = 0.
B. Determine the following as functions of time \(t\).
i. The kinetic energy of the box
ii. The net force acting on the box
iii. The power being delivered to the box
C. Calculate the net work done on the box in the interval \(t\) = 0 to \(t\) = 2 s would be greater than, less than, or equal to the answer in part (C). Justify your answer.
The figure below depicts a roller coaster. Assume the roller coaster starts at a velocity of 0 m/s from 70 m.
A. Determine the velocity of the roller coaster at the top of the loop, 40 m above the ground.
B. Determine the height of the track at point x. It is known that the roller coaster has a velocity of 30 m/s at x.
A marble, m, of mass 100 g is placed onto a platform attached to a spring. The weight of the marble compresses the spring 2 cm, as shown below. A force is then applied to the platform to compress the spring an additional 3 cm. When the force is taken away, the spring decompresses and launches the marble upward. What is the maximum height the marble obtains above the unstretched spring (the spring with no mass on it).
A nonlinear spring is compressed various distances \(x\) , and the force \(F\) required to compress it is measured for each distance. The data are shown in the table below.
Assume that the magnitude of the force applied by the spring is of the form \(F(x) = Ax^2\) .
In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs the cord from a rod and then attaches a variety of weights to the cord. For each weight, the student allows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the table below:
A. Use the data to plot a graph of weight versus length. Sketch a best-fit straight line through the data.
B. Use the best-fit line you sketched in part (A) to determine an experimental value for the spring constant \(k\) of the cord.
The student now attaches an object of unknown mass \(m\) to the cord and holds the object adjacent to the point at which the top of the cord is tied to the rod. When the object is released from rest, it falls 1.5 m before stopping and turning around. Assume that air resistance is negligible.
C. Calculate the value of the unknown mass \(m\) of the object.
i. Calculate how far down the object has fallen at the moment it attains its maximum speed.
ii. Explain why this is the point at which the object has its maximum speed.
iii. Calculate the maximum speed of the object.
A rubber ball of mass \(m\) is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused by the air). The drag force on the ball has a magnitude \(bv^2\) , where \(b\) is a constant drag coefficient and \(v\) is the instantaneous speed of the ball. The drag coefficient \(b\) is directly proportional to the cross-sectional area of the ball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approaches a constant value called the terminal speed.
A. Draw and label all the forces on the ball at some instant before it reaches terminal speed.
B. State whether the magnitude of the acceleration of the ball of mass \(m\) increases, decreases, or remains the same as the ball approaches terminal speed. Explain.
C. Write, but do NOT solve, a differential equation for the instantaneous speed \(v\) of the ball in terms of time \(t\) , the given quantities, and fundamental constants.
D. Determine the terminal speed \(v_t\) in terms of the given quantities and fundamental constants.
E. Determine the energy dissipated by the drag force during the fall if the ball is released at height \(h\) and reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental constants.