A 0.50kg mass is suspended on a spring that stretches 3.0cm.
a. What is the spring constant?
b. What added mass would stretch the spring an additional 2.0cm?
c. What is the change in potential energy when the mass is added?
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weight = m g = 0.50 * 9.8 = 4.9 N
k = 4.9 Newtons / 0.030 meter
b.) .050 meters total x
m g = (4.9 / 0.03) 0.05
so
m = .5/.03*.05 = 0.83 kg total
0.83 - 0.50 = 0.33 Kg additional
c) U = (1/2) k x^2
change = (1/2)(4.9/.03) (.05^2-.03^2)
see your previous post for clues...
a. To find the spring constant, you can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is F = -kx, where F is the force, k is the spring constant, and x is the displacement.
In this case, the mass is suspended, so the gravitational force acting on it is balanced by the force exerted by the spring. The gravitational force can be calculated using the formula F = mg, where m is the mass and g is the acceleration due to gravity.
Since the spring force and gravitational force are equal at equilibrium, we can equate the two formulas:
-kx = mg
Here, m = 0.50 kg and x = 3.0 cm = 0.03 m. Plugging these values into the equation:
-k(0.03) = (0.50)(9.8)
Simplifying the equation:
k = (0.50)(9.8)/(0.03)
Evaluating the expression on the right side:
k ≈ 16.33 N/m
Therefore, the spring constant is approximately 16.33 N/m.
b. To calculate the added mass required to stretch the spring an additional 2.0 cm, we can again use Hooke's Law. Rearranging the formula, we have x = -F/k. Here, x is the displacement, F is the force applied, and k is the spring constant.
Given that the spring has stretched an additional 2.0 cm = 0.02 m, we need to find the force required to cause this displacement. Let's call the unknown mass required to stretch the spring m1.
-F/k = -F1/k = m1g
Where F1 is the force required.
The initial displacement was 0.03 m, and now we want an extra 0.02 m of displacement. So:
m1g = F1/k = (0.50 + m1)g
Rearranging the equation:
0.02g = m1g
Canceling the g:
0.02 = m1
Therefore, the added mass required to stretch the spring an additional 2.0 cm is 0.02 kg, which is equal to 20 grams.
c. The change in potential energy can be calculated using the formula for gravitational potential energy, which is U = mgh. Here, m is the mass, g is the acceleration due to gravity, and h is the change in height.
In this case, the change in height is the change in displacement of the spring, which is an additional 2.0 cm = 0.02 m.
Using the given mass of 0.50 kg:
U = (0.50)(9.8)(0.02)
Evaluating the expression:
U ≈ 0.98 J
Therefore, the change in potential energy when the mass is added is approximately 0.98 Joules.