A horizontal spring attached to a wall has a force constant of 760 N/m. A block

Question

A horizontal spring attached to a wall has a force constant of 760 N/m. A block of mass 1.60 kg is attached to the spring and oscillates freely on a horizontal, frictionless surface as in the figure below. The initial goal of this problem is to find the velocity at the equilibrium point after the block is released. Find the energy stored in the spring when the mass is stretched 6.60 cm from equilibrium and again when the mass passes through equilibrium after being released from rest. x = 6.60 cm J x = 0 cm(d) Write the conservation of energy equation for this situation and solve it for the speed of the mass as it passes equilibrium. Substitute to obtain a numerical value.(e) What is the speed at the halfway point?

Explanation / Answer

energy stored in spring when spring is stretchd by 6.60 cm = 0.5 kx^2 = 0.5 (760)(0.066)^2 = 1.66 J

energy stored in spring is zero when mass passes through equilibrium as complete initial spring energy is converted to kinetic energy of block which is 1.66 J

(d) Initial potential energy = Final kinetic energy at equilibrium position

0.5 kx^2 = 0.5 mvo^2

0.5(760)(0.0660)^2 = 0.5(1.6)(vo^2)

vo =1.4 m/s

(e) Speed at halfway point x'= 0.0660/2 = 0.0330 m

initial potential energy = final kinetic energy + final potential energy

0.5 kx^2 = 0.5 mv^2 + 0.5 kx'^2

0.5(760)(0.0660)^2 = 0.5 (1.6)v^2 + 0.5(760)(0.0330)^2

v= 1.24 m/s

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