9. Work & Energy

(II) A spring has k = 65 N/m . Draw a graph like that in Fig. 7–11 and use it to determine the work needed to stretch the spring from x = 3.0cm to x = 7.5cm , where x = 0 refers to the spring’s unstretched length.

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(II) At the top of a pole vault, an athlete actually can do work pushing on the pole before releasing it. Suppose the pushing force that the pole exerts back on the athlete is given by F(y) = (1.5 x 10² N/m) y- (1.9 x 10² N/m²) y² acting over a displacement from y= 0 (y is vertical) to y = 0.20m . How much work is done on the athlete?

(II) If it requires 5.0 J of work to stretch a particular spring by 2.0 cm from its equilibrium length, how much more work will be required to stretch it an additional 4.0 cm?

(III) A spring of spring constant k and negligible mass is attached to a mass m and the other end of the spring is pulled vertically in order to lift the mass. Find an expression for the amount of work that must be done on the spring before the mass begins to leave the ground.

The force required to compress an “imperfect” horizontal spring (doesn’t follow Hooke’s law) an amount x is given by F = 150x + 12x³ , where x is in meters and F in newtons. If the spring is compressed 2.0 m, what speed will it give to a 3.0-kg ball held against it and then released?

Many cars have “3 mi/h (5 km/h) bumpers” that are designed to compress and rebound elastically without any physical damage at speeds below 5 km/h. If the material of the bumpers permanently deforms after a compression of 1.5 cm, but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of 1050 kg and is tested by driving it into a solid wall?

A simple pendulum consists of a small object of mass m (the “bob”) suspended by a cord of length ℓ (Fig. 7–34) of negligible mass. A force F→ is applied in the horizontal direction (so F→ = Fî ), moving the bob very slowly so the acceleration is essentially zero. (Note that the magnitude of F→ will need to vary with the angle θ that the cord makes with the vertical at any moment.)

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(a) Determine the work done by this force, F→ , to move the pendulum from θ = 0 to θ₀ .