A mass of 50 g is attached to a spring with a spring constant of 1000N/m. The spring is stretched 50 mm from its equilibrium and released. What is the maximum acceleration of the mass?
Well, that's quite a springy situation you've got there! Let's do some calculations to find out the maximum acceleration.
First, let's convert the mass from grams to kilograms. 50 grams is equal to 0.05 kilograms. So now we have a mass of 0.05 kg.
Next, let's convert the displacement from millimeters to meters. 50 mm is equal to 0.05 meters. So now we have a displacement of 0.05 m.
Now we can use Hooke's Law:
F = -kx
F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position. Since we know the values for k and x, we can calculate the force:
F = -1000N/m * 0.05m
F = -50N
Now, let's use Newton's second law, F = ma, to find the maximum acceleration:
-50N = 0.05kg * a
Solving for a, we get:
a = -50N / 0.05kg
a = -1000 m/s²
So the maximum acceleration of the mass is -1000 m/s². But don't worry, it's just a negative sign, nothing to be afraid of!
To find the maximum acceleration of the mass, we can use Hooke's law and the formula for acceleration in simple harmonic motion.
1. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position:
F = -kx
where:
F is the force exerted by the spring,
k is the spring constant (1000 N/m),
x is the displacement from the equilibrium position.
2. Since the mass is attached to the spring, the force exerted by the spring is equal to the force on the mass. Therefore, the maximum force on the mass at the extreme ends of its oscillation will occur when the displacement is maximum.
F = -kx_max
3. The acceleration of the mass can be found using Newton's second law:
F = ma
where:
m is the mass (50 g = 0.05 kg),
a is the acceleration of the mass.
4. Substitute the values into the equation and solve for the acceleration:
-kx_max = ma
-1000 N/m * 0.05 m = 0.05 kg * a
-50 N = 0.05 kg * a
5. Divide both sides of the equation by 0.05 kg to solve for the acceleration:
a = -50 N / 0.05 kg
a ≈ -1000 m/s²
Therefore, the maximum acceleration of the mass is approximately -1000 m/s². The negative sign indicates that the acceleration is directed opposite to the displacement, as required by simple harmonic motion.
To find the maximum acceleration of the mass, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
First, let's convert the displacement from millimeters to meters, as the spring constant is given in Newtons per meter. We have a displacement of 50 mm, which is equal to 0.05 meters.
The force exerted by the spring is given by the equation F = -kx, where F is the force, k is the spring constant, and x is the displacement from equilibrium.
In this case, the mass is not mentioned, so we can assume it has no effect on the maximum acceleration since it cancels out in the equation F = ma.
Now, substituting the given values into the equation, we have:
F = -kx
F = -(1000 N/m)(0.05 m)
F = -50 N
Since acceleration is equal to the net force divided by mass, and the mass is 50 g (0.05 kg), we can calculate the maximum acceleration by dividing the force by the mass:
a = F/m
a = -50 N / 0.05 kg
a = -1000 m/s^2
The negative sign indicates that the acceleration is in the opposite direction of the displacement, which means it is directed towards the equilibrium position. The magnitude of the maximum acceleration is 1000 m/s^2.