A 0.20 kg object, attached to a spring with spring constant k = 10 N/m, is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.080 m. What is its speed at the instant when its displacement is 0.040 m? (Hint: Use conservation of energy.)
49cm/s
A 0.20 kg object, attached to a spring with spring constant k = 10 N/m, is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.080 m. What is its speed at the instant when its displacement is 0.040 …
To find the speed of the object at the given displacement, we can use the principle of conservation of energy. In simple harmonic motion, the total mechanical energy remains constant throughout the motion.
The total mechanical energy of the object can be calculated by summing up the potential energy and kinetic energy at any given point.
1. Potential Energy (PE):
The potential energy of the spring is given by the equation: PE = (1/2)kx^2
Where k is the spring constant and x is the displacement from the equilibrium position.
2. Kinetic Energy (KE):
The kinetic energy of the object is given by the equation: KE = (1/2)mv^2
Where m is the mass of the object and v is its velocity.
At the extreme points of simple harmonic motion, the kinetic energy is zero, and all the energy is in the form of potential energy. At the equilibrium position, the potential energy is zero, and all the energy is in the form of kinetic energy.
Using the given information, let's find the potential energy at the given displacement:
PE = (1/2)kx^2
= (1/2)(10 N/m)(0.040 m)^2
= 0.008 J
Since the total mechanical energy remains constant, the potential energy at the given displacement is equal to the initial mechanical energy of the system.
Next, let's find the initial mechanical energy by considering the amplitude of the motion:
Amplitude (A) = 0.080 m
At the extreme points of simple harmonic motion, the total mechanical energy is equal to the potential energy:
Total Mechanical Energy = PE = (1/2)kA^2
= (1/2)(10 N/m)(0.080 m)^2
= 0.032 J
Since the total mechanical energy remains constant, the initial mechanical energy is equal to the final mechanical energy.
Now, we can express the final mechanical energy in terms of the velocity (v) at the given displacement:
Total Mechanical Energy = PE + KE
0.032 J = 0.008 J + (1/2)mv^2
Simplifying the equation:
0.024 J = (1/2)mv^2
Multiplying both sides by 2 to eliminate the fraction:
0.048 J = mv^2
Now, we can solve for v:
v^2 = 0.048 J / m
v^2 = 0.048 J / 0.20 kg
v^2 = 0.24 m^2/s^2
Taking the square root of both sides:
v = sqrt(0.24 m^2/s^2)
v ≈ 0.49 m/s
Therefore, the speed of the object at the instant when its displacement is 0.040 m is approximately 0.49 m/s.
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Here's another hint: The total energy is
(1/2)k*(Amplitude)^2
When displacement is half the amplitude, potential energy is 1/4 of the total energy. The rest is kinetic.
You should be able to complete the problem