If a car has a suspension system with a force constant of 5.00 times 10 to the fourth newtons per meter, how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 meters?
In this problem statement, we’re told the force constant, which we’ll call 𝑘, is 5.00 times 10 to the fourth newtons per meter. And we’re also told the maximum displacement of oscillation, which we’ll call 𝑥 sub max, is 0.0750 meters. We’re asked to solve for the energy the car’s shocks must remove. We’ll call this energy capital 𝐸. Because these shocks are made of springs, the energy that the shocks must remove is in fact spring energy.
So to solve for this value, let’s recall the equation for the potential energy stored in a spring. The energy stored in a spring is equal to one-half the spring constant, 𝑘, multiplied by its displacement from equilibrium, 𝑥 squared. If we apply this equation to our scenario, the energy that the car’s shocks need to remove, capital 𝐸, is equal to half the force constant multiplied by 𝑥 max squared.
We’re given 𝑘 and 𝑥 sub max. 𝑘 is 5.00 times 10 to the fourth newtons per meter, and 𝑥 max is 0.0750 meters. Plugging these values into our equation, we find that the energy that the car’s shocks must remove is equal to 141 joules, to three significant figures. That’s how much energy is needed to dampen this oscillation.