A 0.5 kg air-track car is attached to the end of a horizontal spring of constant k = 20 N/m. The car is displaced 15 cm from its equilibrium point and released.
a) What is the car's maximum speed?
b) What is the car's maximum acceleration?
c) What is the frequency f of the car's oscillation?
I need help please. What equations should I use for each problems?
To solve these problems, we can use the equations related to the motion of a mass-spring system.
a) To find the car's maximum speed, we need to use the concept of conservation of mechanical energy. The maximum speed occurs when all the potential energy stored in the spring is converted into kinetic energy. The equation we can use is:
(1/2)mv^2 = (1/2)kx^2
where m is the mass of the car (0.5 kg), v is the maximum speed we are trying to find, k is the spring constant (20 N/m), and x is the displacement from equilibrium (0.15 m).
Substituting the given values into the equation, we can solve for v:
(1/2)(0.5 kg)v^2 = (1/2)(20 N/m)(0.15 m)^2
Simplifying and solving for v, we get:
v = √[(20 N/m)(0.15 m)^2 / (0.5 kg)]
b) To find the car's maximum acceleration, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from equilibrium. The equation we can use is:
F = -kx
where F is the force exerted by the spring, k is the spring constant (20 N/m), and x is the displacement from equilibrium (0.15 m).
Since the mass of the car is 0.5 kg, we can use Newton's second law, F = ma, to relate force to acceleration:
ma = -kx
Rearranging the equation, we get:
a = (-k/m)x
Substituting the given values into the equation, we can solve for a:
a = (-20 N/m / 0.5 kg)(0.15 m)
c) To find the frequency f of the car's oscillation, we can use the equation:
f = (1 / 2π) √(k / m)
where f is the frequency, k is the spring constant (20 N/m), and m is the mass of the car (0.5 kg).
Substituting the given values into the equation, we can solve for f.