8. A spring is compressed a distance of 0.15 m from its rest position and held in place while a ball of mass 0.150 kg is placed at its end. When the spring is released, the ball leaves the spring traveling at 9.0 m/s. What is the spring constant?
540 N/m
450 N/m
60 N/m
225 N/m
122.5 N/m
I got 225 N/m is this correct?
(1/2) m v^2 = (1/2) k x^2
.15 (81) = k (.0225)
k = 540 N/m
To find the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, this can be represented as:
F = k * x
Where:
F is the force exerted by the spring (in Newtons),
k is the spring constant (in Newtons per meter),
x is the displacement from the equilibrium position (in meters).
In this problem, we know the displacement x (0.15m) and the mass of the ball (0.150kg), and we need to find the spring constant k.
First, let's find the force exerted by the spring. We can use Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration. In this case, the acceleration can be calculated using the final velocity of the ball:
v = u + at
Where:
v is the final velocity (9.0m/s),
u is the initial velocity (0m/s),
a is the acceleration (unknown),
t is the time taken (unknown).
Since the ball starts from rest, the initial velocity (u) is 0m/s. Therefore, the equation simplifies to:
v = at
Solving for the acceleration (a):
a = v / t
Now, let's consider the motion of the ball when it is released from the spring. Initially, only the spring exerts a force on the ball, causing it to accelerate. However, as the ball moves, the spring force decreases until it is balanced by the force of gravity (mg), which causes the ball to reach a constant velocity.
At the equilibrium position, both forces cancel each other out:
k * x = mg
Rearranging the equation to solve for the spring constant (k):
k = mg / x
Now we have all the information needed to calculate the spring constant.
Step 1: Calculate the acceleration (a):
a = v / t
In the question, the final velocity of the ball (v) is 9.0m/s, but the time taken (t) is not given. Therefore, we need additional information to solve for time.
Step 2: Calculate the spring constant (k):
k = mg / x
Here, the mass of the ball (m) is given as 0.150kg, and the displacement of the spring (x) is given as 0.15m.
Using the value of acceleration and substituting the mass and displacement values into the formula, we can now calculate the spring constant.
Now, let's compute the spring constant using the given values:
k = mg / x
= (0.150kg)(9.8m/s²) / 0.15m
≈ 0.980 N / 0.15m
≈ 6.53 N/m
The closest option to this value is 60 N/m, so the spring constant is approximately 60 N/m.