11.5✶✶ (a) Find the normal frequencies, w₁ and w₂, for the two carts shown in Figure 11.15, assuming that m₁ = = m2 and k₁ = k2. (b) Find and describe the motion for each of the normal modes in turn. Problems for Chapter 11 449 k₂ k₁ m ma X1 x2 Figure 11.15 Problems 11.5 and 11.6. 11.6** Answer the same questions as in Problem 11.5 but for the case that m₁ = m2 and k₁ = 3k2/2. (Write k₁ = 3k and k₂ = 2k.) Explain the motion in the two normal modes.

A 4.00 [kg] frictionless block is attached to an ideal spring of force constant 400.[N/m]. At ? = 0 the block has velocity -5.00 [m/s] and displacement 0.200 [m].  Now solve for this:  A. The amplitude of the oscillation B. The position-time and velocity-time functions of the spring-mass system Need help thank you.

The position 7 (in meters) of a particle of mass m (in kilograms) is described as two perpendicular oscillations that are out of phase with each other: 7 = rēi + yế2 = cos(wt)ěi + cos(wt + ø)ē2, where o is the constant phase angle difference. a. Show that the position 7 of the particle satisfies the harmonic oscillator equation -w?r. dt? Compute the particle's velocity v = dr/dt, dot product 7 · v, and angular momentum L = mĩ x v. Is the angular momentum constant in both magnitude and direction? b. Set w = 2 rad / s and ø = T/3 rad. Make a table with the following columns: t, x, y, Væ, Vy, and7· v. Put the units beside their respective variables and enclose the units inside the parentheses. Set the initial time to be to = 0 and use the spreadsheet to compute the values of xo, Yo, VOx, VOy, and ro · vo Write down the spreadsheet formula for the first xo, Yo, VOz, VOys and 7o - vo , assuming that the initial time to is spreadsheet cell A2.

A cart of mass m = 1.6 kg placed on a frictionless horizontal surface and connected to a spring with spring constant 6 N/m. The damping force strength is given by b = 255 g/s. The cart is pulled away 16.5 cm and released. k m 'b a. Calculate the time required for the amplitude of the resulting oscillations to fall to 1/7 of its initial value. 24.4 b. How many oscillations are made by the cart in this time? 7.51

5 2 8. The power output, P, in Watts of an electric generator varies sinusoidally over the course of a day, with a period of a day. The output is a minimum of 200 watts at midnight and the maximum output is 800 watts. Let t be the number of hours since midnight. What is the average power output? What's the amplitude? Draw the graph of P against t over two days. Find a formula for P n terms of t. Is it easier to modify a sine formula, or a cosine formula?

A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at rest in a position yi such that the spring is at its rest length. The object is then released from y; and oscillated up and down, with its lowest position being 10 cm below Yi. a. What is the frequency of oscillation? b. What is the speed (in m/s) of the object when it is 8 cm below the initial position? c. An object of mass 340 g is attached to the first object, after which the system oscillated with half the original frequency. What is the mass (in grams) of the first object? d. How far below (in cm) Yi is the new equilibrium (rest) position with both objects attached to the spring?

There is a particle of mass m in the following potential. x a How many useful modes are there? How?

Q.1 Consider the following equation of motion 2x + cx + 800x = 50 sin(10t) find the solution of the steady state response.

the maximum energy of a vibrating mass iss 3.00x10^-2 joules. the amplitude of oscillation is 2.50mm with a frequency of 200Hz. a. what is the mass of the object? b. what is the maximum acceleration of the object? c. if the mass affects the vibration frequency (like that of a mass and spring system), what should the mass be to double the maximum energy? assume that the oscillation amplitude is not affected.

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