A Mass Attached To A Spring Oscillates With The Following Po ✓ Solved

A mass attached to a spring oscillates with the following posit

How many oscillations does the mass make per second? (a) 0.79 (b) 5 (c) 31.4 (d) 1.59 (e) . For the mass in the previous problem, what is its speed at t = 7 s? (a) 27 cm/s (b) 1.61 cm/s (c) 13.88 cm/s (d) 9.64 cm/s (e) 0 cm/s. A banked curve on a race track is designed to operate without friction for a car traveling at a speed of 47 m/s. A driver goes through the curve at a speed of 31 m/s. In which direction does the friction force on the car tires point (if there is a friction force)? (a) Banked tracks do not require friction. (b) The friction force points radially inward towards the center of the circular motion (the friction force direction is parallel to the ground). (c) The friction force is parallel to the banked track and points up the slope of the banked track. (d) The friction force is parallel to the banked track and points down the slope of the banked track. (e) The friction force points in the direction of the normal force. The density of lead is 11.36 g/cm3 and the mass number of lead is 208 g/mol. Calculate the interatomic spacing of lead (Avogadro’s number = 6.02 x 1023). (a) 10.9 nm (b) 1.02 nm (c) 0.31 nm (d) 9.77 nm (e) 1194 nm. What is the total energy of a proton of mass, m= 1.67 x 10-27 kg, moving at 0.99c. (a) 7.4 x 10-11 J (b) 3.55 x 10-18 J (c) 1.5 x 10-10 J (d) 8.2 x 10-28 J (e) 1.07 x 10-9 J.

All work must be shown for full credit. Answers must have appropriate units. A steel beam of mass 250 kg and length 8 m is attached to a vertical wall by a hinge. The beam is supported in a horizontal position by a cable that attaches to the beam at a point 6 m from the wall. The cable makes an angle of 35o to the beam as shown in the figure below. (a) Calculate the tension in the cable when Walter (mass = 110 kg) is at the end of the beam. Carefully show your reasoning. (b) Determine the horizontal and vertical force components that must be applied to the beam at the hinge point where it meets the wall. On a hot wheels track, the car of mass 38 g starts at a height of 1.2 m above the level of the table. The loop has a diameter of 0.38 m. Assume the track is frictionless apart from a rough patch of length x with coefficient of kinetic friction µk=0.91. (a) Calculate the speed of the car at the top of the loop. (b) Determine the normal force on the car when it is at the top of the loop. (c) Determine the speed and the normal force on the car at the bottom of the loop after it has gone over the top. (d) Find the minimum distance x of the rough patch required to just bring the car to a stop. A metal wire of length 3 m and diameter 4 mm is attached to the ceiling. The wire extends in length by 6 mm when an 80 kg mass is hung from the wire. (a) Assuming the wire acts like a spring, determine the effective spring constant, keff, for the wire. (b) The metal from which the wire is made has an interatomic spacing of d=0.14 nm. Using the ball and interatomic spring model of the atoms in the wire, determine the number of parallel springs, Nparallel, and the number of springs in series, Nseries, for the wire. (c) Determine the spring constant for the interatomic bonds between atoms. (d) Determine Young’s modulus for the metal.

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The oscillation of mass attached to a spring is a fundamental concept in physics, particularly in mechanics and wave theory. The position function of the mass, given by y = 0.5 cos(5t + 2), illustrates how the mass moves over time. To calculate the number of oscillations made per second, we can analyze the position function closely. The general form of a cosine function is given by y = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. In this case, ω = 5. This means that the frequency f, which is the number of oscillations per second, can be derived from the angular frequency using the formula f = ω / (2π). Thus, substituting ω = 5 into the equation yields f = 5 / (2π) ≈ 0.796 oscillations per second. Therefore, the correct answer to the first part of the multiple-choice question is (a) 0.79.

Following this, we need to evaluate the speed of the mass at time t = 7 seconds using the velocity function, which is the first derivative of the position function. The first derivative of y = 0.5 cos(5t + 2) is given by v(t) = -0.5 * 5 sin(5t + 2) = -2.5 sin(5t + 2). Plugging t = 7 seconds into this velocity equation allows us to compute the speed. Performing the necessary calculations will give us the specific speed at that moment.

Regarding the friction force on a vehicle navigating a banked curve on a racetrack, we consider the definitions of motion on curved paths. The original question proposed that the ideal speed that negates the need for friction is 47 m/s, but the car encounters a speed of 31 m/s. Thus, a friction force will be required. It will point radially inward toward the center of the circular path because friction helps provide the necessary centripetal force in scenarios where speed is lower than designed. Hence, the answer is (b): The friction force points radially inward towards the center of the circular motion.

Next, when tasked with calculating the interatomic spacing of lead, which involves density and the molar mass of lead, we utilize the formula linking these quantities through Avogadro's number. We calculate the volume of one mole of lead by dividing its mass by its density. Consequently, using this volume and Avogadro's number allows us to derive the interatomic spacing. The details of the calculation will guide us to one of the multiple-choice answers.

Lastly, in determining the total energy of a proton moving at relativistic speeds, we harness the complete energy formula. The total energy E is expressed as E = γmc², where γ is the Lorentz factor which can be calculated based on the velocity provided. By implementing the mass of the proton and its velocity of 0.99c into our formulas, we reach the correct answer aligned with the given choices.

Moving on to practical problems, we must tackle the issue of a steel beam subjected to forces and torques while raised horizontally. The center of mass approach helps calculate the tension in the supporting cable as well as the forces at the hinge. Using torque equilibrium about the hinge point gives necessary equations to solve for the tension, while resolving forces horizontally and vertically helps determine other required forces.

Next, we consider the car on a frictionless hot wheels track. Amongst criteria established in the scenario, calculating velocity at the loop's apex involves energy conservation principles, equating potential energy loss to kinetic energy gain. Utilizing known values allows us to compute speeds and forces elsewhere in the loop, and estimating the lengths of rough patches needed to halt the car additionally uses work-energy principles united with the friction consideration.

Finally, for the metal wire, the Young's modulus reflects the wire's behavior under tension, and aspects of the structure as parallel and series springs together summarize the internal forces at play. Inclusively, the number of springs in parallel and in series is derived from geometrical properties and the interatomic spacing provided, leading to an understanding of the spring constants involved.

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