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Daniel B.
Teacher for 15 years
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Physics (Newtonian Mechanics)
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Question:

A common carnival ride has a large cylindrical walled room that spins about a central axis. The riders are pressed against the wall of the cylinder at which point the floor drops down leaving the rider suspended by against the wall of the rotating cylinder. if the coefficient of friction between the rider and the wall is 0.25, and the radius of the cylinder is 5.0 meters, what frequency must the ride spin so that the riders don't fall?

Daniel B.
Answer:

The coefficient of friction is μ=0.25 and the radius of the spinning cylinder is r=5.0 meters The friction responsible for suspending the rider is given by f=μN, where N is the normal force from the surface on to the rider. The free body diagram of the rider shows the weight of the rider (mg) must be balanced by the friction force. Thus μN=mg N=mg/μ Also, the centripetal force is supplied by the Normal force (N) of the surface pushing the rider to move in a circle, which is given by the equation N=mv^2/r Equating these two results yields mg/μ=(mv^2)/r v=√(rg/μ) The speed of an object moving in a circle is related to the frequency of rotation (f) by v=2πrf Equating the two equations for speed yields f= √(rg/μ)/2πr Now, substituting for the radius, coefficient of friction, acceleration of gravity constant (g=9.8m/s^2) and π yields the answer f=0.45 rev/s

Pre-Calculus
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Question:

Solve the following quadratic equation: 2x^{2} - (x +2)(x -3) = 12

Daniel B.
Answer:

First, multiply the binomials to leave only monomial terms in the equation, and collect all terms on the left side 2x^{2} - (x^{2} - 3x +2x - 6) - 12 = 0 Combine like terms, and write in standard form (2x^{2} - x^{2} ) + (-3x + 2x) + (6 - 12) = 0 x^{2} - x - 6 = 0 Now factor to obtain the roots (x - 3)(x + 2) x = 3 and x = -2 are the solutions.

Physics
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Question:

A car slides 40 meters to a stop on wet pavement after the driver slams on the breaks, eventually hitting another car in an intersection. The police officer investigating the crash approximates the coefficient of the sliding friction to be about 0.40. Determine if the driver was exceeding the speed limit of 30 mph (13.4 m/s).

Daniel B.
Answer:

Using the relationship Work = Change in Kinetic Energy, the friction responsible for slowing the car down does negative work on the car. The kinetic energy (KE) is thus reduced to zero (the car stops), so that setting the two calculations equal to each other yields: friction x distance = final KE - initial KE (0.4) x (normal force on car) x (slide distance) = 0 - (0.5)(mass of car)(initial speed of car)^2 (0.4) x (mass of car) x (accel. of gravity) x (40 m) = - (0.5)(mass of car)(initial speed of car)^2 Note: Canceling the mass from each side of equation reminds us that the mass of the car is not required for a solution. Solving for the initial speed of the car yields a speed of v = 12.5 m/s, which means the driver was indeed below the 30 mph limit.

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