after a spring is compressed 0.2 meters from its uncompressed length it exerts a 20 newton force on a 0.4 kg block resting on a horizontal frictionless surface. after it is released the block will reach a maximum velocity of
a. 2.1 m/s
b. 3.2 m/s
c. 6.3 m/s
d. 10.0 m/s
e. 100 m/s
spring constant ... k = 20 N / 0.2 m
stored energy = 1/2 k x^2 = 1/2 * 100 * 0.2^2 Joules
the K.E. of the block equals the stored energy in the spring
1/2 m v^2 = 2 J
v^2 = 4 J / 0.4 kg
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of the system remains constant if no external forces are acting on it.
The mechanical energy in this system is composed of the potential energy stored in the compressed spring and the kinetic energy of the moving block.
1. First, we need to calculate the potential energy stored in the compressed spring. The potential energy in a spring is given by the formula:
Potential Energy = (1/2) * k * x^2
Where k is the spring constant and x is the displacement of the spring from its uncompressed length.
In this case, the displacement of the spring is 0.2 meters, and we need to find the spring constant.
2. The force exerted by the spring is given by Hooke's Law:
Force = k * x
We know that the force exerted by the spring is 20 Newtons when the displacement is 0.2 meters.
Substituting the known values into the equation, we can solve for the spring constant (k).
3. With the spring constant, we can calculate the potential energy stored in the spring using the formula from step 1.
4. The maximum velocity of the block can be found by equating the potential energy to the kinetic energy of the block:
Potential Energy = Kinetic Energy
(1/2) * m * v^2 = Potential Energy
Where m is the mass of the block and v is the velocity of the block.
5. Rearranging the equation, we can solve for the velocity (v).
Now, let's calculate the maximum velocity of the block:
Given:
Displacement, x = 0.2 meters
Force, F = 20 Newtons
Mass, m = 0.4 kg
Step 1: Calculating the spring constant (k)
Force = k * x
20 N = k * 0.2 m
k = 20 N / 0.2 m
k = 100 N/m
Step 2: Calculating the potential energy stored in the spring
Potential Energy = (1/2) * k * x^2
Potential Energy = (1/2) * 100 N/m * (0.2 m)^2
Potential Energy = 2 Joules
Step 3: Calculating the maximum velocity of the block
Potential Energy = Kinetic Energy
(1/2) * m * v^2 = Potential Energy
(1/2) * 0.4 kg * v^2 = 2 J
v^2 = (2 J) / (0.4 kg * (1/2))
v^2 = 10 J/kg
v = sqrt(10 J/kg)
v ≈ 3.16 m/s
Therefore, the maximum velocity of the block is approximately 3.2 m/s.
So, the correct answer is b. 3.2 m/s.
To determine the maximum velocity of the block, we can use the conservation of mechanical energy. When the block is at maximum velocity, all of the potential energy stored in the compressed spring will be converted into kinetic energy.
First, let's determine the potential energy stored in the spring when it is compressed by 0.2 meters. The potential energy of a spring can be calculated using the formula:
Potential energy (PE) = (1/2) * k * x^2
Where k is the spring constant and x is the displacement from the equilibrium position.
Next, let's determine the spring constant for the given spring. The spring constant can be calculated using Hooke's law:
Force (F) = k * x
Rearranging the formula, we can solve for k:
k = F / x
Now, we can calculate the spring constant:
k = 20 N / 0.2 m = 100 N/m
With the spring constant determined, we can find the potential energy stored in the spring:
PE = (1/2) * 100 N/m * (0.2 m)^2 = 2 J
Since the potential energy is converted entirely into kinetic energy at maximum velocity, we can equate them:
Kinetic energy (KE) = PE
Using the formula for kinetic energy:
KE = (1/2) * m * v^2
Where m is the mass of the block and v is the velocity. Rearranging the formula, we can solve for velocity:
v = √(2 * PE / m)
Substituting the values:
v = √(2 * 2 J / 0.4 kg) = √(10 m^2/s^2) ≈ 3.16 m/s
Therefore, the maximum velocity of the block is approximately 3.16 m/s, which is closest to option (b) 3.2 m/s.