A 2.60-kg block is attached to a horizontal spring that has a spring constant of 203 N/m. At the instant when the displacement of the spring from its unstrained length is -0.115 m, what is the acceleration a of the object? In your answer, be sure to include the proper plus or minus sign.
To find the acceleration of the object, we can use Hooke's Law and the equation of motion.
Step 1: Determine the force exerted by the spring.
According to Hooke's Law, the force exerted by a spring is given by:
F = -kx
where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the displacement (x) is given as -0.115 m, and the spring constant (k) is 203 N/m. Therefore, we can calculate the force exerted by the spring:
F = -203 N/m * (-0.115 m)
F = 23.345 N
Note that the negative sign indicates that the force is in the opposite direction of the displacement.
Step 2: Calculate the acceleration.
To find the acceleration, we need to apply Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
F_net = m * a
In this case, the force exerted by the spring is the net force acting on the object. Therefore, we can substitute the force value (23.345 N) and the mass of the object (2.60 kg) into the equation:
23.345 N = 2.60 kg * a
Now, solve the equation for acceleration:
a = 23.345 N / 2.60 kg
a ≈ 8.96 m/s²
The acceleration of the object is approximately 8.96 m/s². Note that the acceleration is positive, indicating that it is in the same direction as the force exerted by the spring.
To find the acceleration of the object, we can use Hooke's Law and Newton's second law.
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, and x is the displacement from the equilibrium position.
Newton's second law states that the force acting on an object is equal to its mass times its acceleration:
F = ma
We can equate these two equations:
ma = -kx
Rearranging the equation, we can solve for acceleration:
a = -kx/m
Given:
m = 2.60 kg (mass of the block)
k = 203 N/m (spring constant)
x = -0.115 m (displacement)
Substitute the given values into the equation:
a = -(203 N/m)(-0.115 m) / 2.60 kg
Simplifying the equation:
a = 2.2965 N / 2.60 kg
a ≈ 0.8839 m/s²
Therefore, the acceleration of the object is approximately 0.8839 m/s².
Wb = 2.6kg * 9.8N/kg = 25.48 N.
203N/1m = F/-0.115
F=-0.115*203=-23.35 N.
a = F/m = -23.35/2.60 = -8.98 m/s^2.