A 100 g blob is attached to a vertical spring and the spring stretched 4 cm from its original length.calculate the spring constant, k and period, t

m = .100 Kg

x = .04 meter
F = m g = 9.81(.1) = .981 N
k = F/x = .981/.04 = 24.5 Newtons/meter

let y = A sin 2 pi t/T
F = -ky = -kA sin 2 pi t/T

v = (2 pi /T)A cos 2 pi t/T
a = - (2 pi/T)^2 A sin 2 pi t/T
F = m a
so
k y = m(2 pi /T )^2 y
2 pi/T = sqrt k/m

T = 2 pi sqrt (m/k)
= 2 pi sqrt (.1/24.5)
= 0.401 seconds

To calculate the spring constant, k, and period, t, we can use Hooke's Law and the formula for the period of a mass-spring system.

1. Calculation of the spring constant, k:
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its original length. Mathematically, it can be represented as:
F = -kx
where F is the force exerted by the spring, k is the spring constant, and x is the displacement from its original length.

In this case, we have a 100 g (0.1 kg) blob attached to the spring, and the spring is stretched 4 cm (0.04 m). The force exerted by the spring can be calculated as the weight of the blob:
F = mg
where m is the mass and g is the acceleration due to gravity.

In this case, F = 0.1 kg * 9.8 m/s^2 = 0.98 N.

Substituting the values into Hooke's Law, we have:
0.98 N = -k * 0.04 m

Solving for k, we get:
k = -0.98 N / 0.04 m
k = -24.5 N/m (Note: The negative sign indicates that the force exerted by the spring is in the opposite direction to the displacement)

Therefore, the spring constant is 24.5 N/m.

2. Calculation of the period, t:
The period of a mass-spring system can be calculated using the following formula:
T = 2π√(m/k)
where T is the period, m is the mass, and k is the spring constant.

Substituting the values into the formula, we have:
T = 2π√(0.1 kg / 24.5 N/m)

Simplifying the equation, we get:
T = 2π√(0.00408 kg⋅m/N) (Note: 1 N = 1 kg⋅m/s^2)

Evaluating the expression, we find:
T ≈ 2.027 s

Therefore, the period is approximately 2.027 seconds.

To calculate the spring constant (k) and period (t) of a vertical spring with a 100 g blob attached to it and stretched 4 cm from its original length, you can use the following steps:

Step 1: Convert the mass of the blob from grams to kilograms.
- Mass in kilograms = Mass in grams / 1000
- Mass in kilograms = 100 g / 1000 = 0.1 kg

Step 2: Calculate the gravitational force acting on the blob.
- Gravitational force = Mass * Acceleration due to gravity (g)
- Gravitational force = 0.1 kg * 9.8 m/s^2 = 0.98 N

Step 3: Determine the spring constant (k) using Hooke's Law.
- Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position.
- Hooke's Law equation: F = -k * x
where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
- In this case, the force exerted by the spring is equal to the gravitational force acting on the blob.
- So, we can write the equation as: 0.98 N = -k * 0.04 m
- Rearranging the equation, we get: k = -0.98 N / 0.04 m
- Therefore, the spring constant (k) is equal to 24.5 N/m.

Step 4: Calculate the period (t) of the vertical spring.
- Period (t) is the time it takes for one complete oscillation of the spring, or the time for the blob to move up and down once.
- The formula to calculate the period of a spring is given by: t = 2π * √(m / k)
where t is the period, π is pi (approximately 3.14159), m is the mass, and k is the spring constant.
- Plugging in the values, we get: t = 2π * √(0.1 kg / 24.5 N/m)
- Calculating further, we have: t = 2π * √0.00408163 s^2/kg
- Taking the square root, we get: t = 0.0906 s (rounded to four decimal places)
- Therefore, the period (t) of the vertical spring is approximately 0.0906 seconds.