9. Work & Energy

A $6.0$-kg box moving at $3.0$ m/s on a horizontal, frictionless surface runs into a light spring of force constant $75$ N/cm. Use the work–energy theorem to find the maximum compression of the spring.

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a $16.0$-cm strip of the donated aorta reveal that it stretches $3.75$ cm when a $1.50$-N pull is exerted on it. If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is $1.14$ cm, what is the greatest force it will be able to exert there?

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a $16.0$-cm strip of the donated aorta reveal that it stretches $3.75$ cm when a $1.50$-N pull is exerted on it. What is the force constant of this strip of aortal material?

(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.

(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 $F\left(y\right)=(1.5\times {10}^2\text{N/m})y-(1.9\times {10}^2{\text{N/m}}^2)y^2$ 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 50 g rock is placed in a slingshot and the rubber band is stretched. The magnitude of the force of the rubber band on the rock is shown by the graph in FIGURE P9.56. The rubber band is stretched 30 cm and then released. What is the speed of the rock?

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 $\overrightarrow{F}$ is applied in the horizontal direction (so $\overrightarrow{F}$ = Fî ), moving the bob very slowly so the acceleration is essentially zero. (Note that the magnitude of $\overrightarrow{F}$ will need to vary with the angle θ that the cord makes with the vertical at any moment.) Determine the work done by this force, $\overrightarrow{F}$, to move the pendulum from θ = 0 to θ₀.

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