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When an object is displaced by an amount x from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of x. In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as
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Physics for Scientists and Engineers
- A light spring with spring constant 1 200 N/m is hung from an elevated support. From its lower end hangs a second light spring, which has spring constant 1 800 N/m. An object of mass 1.50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.arrow_forwardConsider a block of mass 0.200 kg attached to a spring of spring constant 100 N/m. The block is placed on a frictionless table, and the other end of the spring is attached to the wall so that the spring is level with the table. The block is then pushed in so that the spring is compressed by 10.0 cm. Find the speed of the block as it crosses (a) the point when the spring is not stretched, (b) 5.00 cm to the left of point in (a), and (c) 5.00 cm to the right of point in (a).arrow_forwardA horizontal spring attached to a wall has a force constant of k = 850 N/m. A block of mass m = 1.00 kg is attached to the spring and rests on a frictionless, horizontal surface as in Figure P7.55. (a) The block is pulled to a position xi = 6.00 cm from equilibrium and released. Find the elastic potential energy stored in the spring when the block is 6.00 cm from equilibrium and when the block passes through equilibrium. (b) Find the speed of the block as it passes through the equilibrium point. (c) What is the speed of the block when it is at a position xi/2 = 3.00 cm? (d) Why isnt the answer to part (c) half the answer to part (b)? Figure P7.55arrow_forward
- A block of mass 0.250 kg is placed on top of a light, vertical spring of force constant 5 000 N/m and pushed downward so that the spring is compressed by 0.100 m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise?arrow_forwardConsider a linear spring, as in Figure 7.7(a), with mass M uniformly distributed along its length. The left end of the spring is fixed, but the right end, at the equilibrium position x=0 , is moving with speed v in the x-direction. What is the total kinetic energy of the spring? (Hint: First express the kinetic energy of an infinitesimal element of the spring dm in terms of the total mass, equilibrium length, speed of the right-hand end, and position along the spring; then integrate.)arrow_forwardRubber tends to be nonlinear as an elastic material. Suppose a particular rubber band exerts a restoring force given by Fx(x) = Ax Bx2, where the empirical constants are A = 14 N/m and B = 3.3 N/m2 so that Fx is in newtons when x is in meters. Calculate the change in elastic potential energy of the rubber band when an external force stretches it from x = 0 to x = 0.20 m.arrow_forward
- A block of mass m = 0.250 kg is pressed against a spring resting on the bottom of a plane inclined an angle = 45.0 to the horizontal. The spring, which has a force constant of 955 N/m, is compressed a distance of 8.00 cm, and the block is released from rest. Consider the total energy of the springblockEarth system. a. What is the total distance the block moves from its initial position if the incline is frictionless? b. What is the total distance the block moves from its initial position if the coefficient of kinetic friction between the incline and the block is 0.330?arrow_forwardConsider the data for a block of mass m = 0.250 kg given in Table P16.59. Friction is negligible. a. What is the mechanical energy of the blockspring system? b. Write expressions for the kinetic and potential energies as functions of time. c. Plot the kinetic energy, potential energy, and mechanical energy as functions of time on the same set of axes. Problems 5965 are grouped. 59. G Table P16.59 gives the position of a block connected to a horizontal spring at several times. Sketch a motion diagram for the block. Table P16.59arrow_forwardProve that when the average is taken with respect to position over one cycle, the average potential energy equals kA^2/6 and the average kinetic energy equals kA^2/3.arrow_forward
- A particle moves in one dimension (along the x-axis), where the potential energy is given by the function U(x) shown below. The constants are A = 2.00 J/m2 and B = 4.00 J/m. The total energy of the particle is E = 8.00 J. U(x) = Ax2 + Bx (a) What is the equilibrium position of the particle (x, in units of meters)? Is the equilibrium position stable or unstable? (b) What range of positions is the particle restricted to? Give both xmin (the smallest possible position) and xmax (the largest possible position), in units of meters. Hint: the particle is restricted to move in the region where E ≥ U(x).arrow_forwardThe figure shows a track made up of two sections and located in a vertical plane. Section ab is straight and horizontal and section bc is shaped like a 1/4 circumference of radius R = 5 m and the center at point d. The track is smooth except for a piece in the straight section of unknown length L, where μ = 7/010 is the coefficient of dynamic friction. At one end of the track there is a spring of elastic constant k = 800 N / m, point e is the equilibrium point of the spring. A particle of mass m = 2 kg starts from rest at point 1 where the spring is compressing a length of x1 = 4/10 m. The particle is not attached to the spring, it reaches point 2 and then returns. a. Find the modulus of the normal force due to the track at point 2. b. Find the length L c. Upon return, the particle compresses the spring again. Find the new maximum understanding of the springarrow_forwardA block of mass m = 1.9 kg is attached to a spring of force constant 500 N/m. The assembly lies on a horizontal frictionless surface. The block is pulled 7.0 cm to the right of equilibrium. What was the work required to stretch the spring?arrow_forward
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