When a 0.101 kg mass is suspended at rest from a certain spring, the spring stretches 3.80 cm. Find the instantaneous acceleration of the mass when it is raised 5.90 cm, compressing the spring 2.10 cm.
Yes
w=kx
.101g=k*3.80
k=.101g/3.80
force of compression=
= (.101g/.0389)(.0590-.0380)
figure that out, then
using f=ma or a=f/m
and we have to add g because the object is going down,
a=g+(.101g/.0389)(.0590-.0380) /.101g)
To find the instantaneous acceleration of the mass, we need to use Hooke's Law and the equations of motion. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Let's break down the problem into a step-by-step solution:
Step 1: Find the spring constant (k):
The spring constant represents the stiffness of the spring and is given by:
k = F / x
where F is the force exerted by the spring and x is the displacement of the spring.
In this case, when the 0.101 kg mass is suspended at rest, the spring stretches 3.80 cm (or 0.038 m). The force exerted by the spring is given by the weight of the mass, which can be calculated as:
F = m * g
where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values, we get:
F = 0.101 kg * 9.8 m/s^2 = 0.9918 N
Now, we can calculate the spring constant:
k = 0.9918 N / 0.038 m ≈ 26.11 N/m
Step 2: Find the restoring force (F) when the spring is compressed by 2.10 cm (or 0.021 m):
The restoring force exerted by the spring can be calculated using Hooke's Law:
F = k * x
where k is the spring constant and x is the displacement of the spring.
Substituting the values, we get:
F = 26.11 N/m * (-0.021 m) = -0.54831 N
Note: The negative sign indicates that the force is in the opposite direction of the displacement.
Step 3: Find the acceleration (a) using Newton's second law:
Newton's second law states that the acceleration of an object is equal to the net force acting on it divided by its mass:
a = F_net / m
In this case, the net force (F_net) is equal to the restoring force exerted by the spring since there are no other forces acting on the object.
Substituting the values, we get:
a = -0.54831 N / 0.101 kg ≈ -5.428 m/s^2
The negative sign indicates that the acceleration is in the opposite direction of the displacement.
Therefore, the instantaneous acceleration of the mass when it is raised 5.90 cm and compresses the spring 2.10 cm is approximately -5.428 m/s^2.