a block with a mass of 3.7 kg slides with a speed of 2.2 m/s on a frictionless surface. the block runs into a stationary spring and compresses it a certain distance before coming to rest. what is the compression distances, given that the spring has a spring constant of 3200 N/m?
To find the compression distance of the spring, we can use the principle of conservation of mechanical energy.
The initial kinetic energy of the block is given by the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Plugging in the values given:
Kinetic Energy = (1/2) * 3.7 kg * (2.2 m/s)^2
Next, we need to find the potential energy stored in the spring when it is fully compressed. The potential energy stored in a spring is given by the formula:
Potential Energy = (1/2) * spring constant * compression^2
Plugging in the values given:
Potential Energy = (1/2) * 3200 N/m * compression^2
According to the principle of conservation of mechanical energy, the initial kinetic energy of the block is equal to the potential energy stored in the spring when it is fully compressed:
(1/2) * 3.7 kg * (2.2 m/s)^2 = (1/2) * 3200 N/m * compression^2
Simplifying the equation:
3.7 kg * (2.2 m/s)^2 = 3200 N/m * compression^2
Rearranging the equation to solve for compression:
compression^2 = (3.7 kg * (2.2 m/s)^2) / (3200 N/m)
compression^2 = 13.42 / 3200
compression^2 = 0.00419375
Taking the square root of both sides to find the compression:
compression = √0.00419375
compression ≈ 0.0647 m or 6.47 cm
Therefore, the compression distance of the spring is approximately 0.0647 meters or 6.47 centimeters.
To find the compression distance of the spring, we can use the principle of conservation of mechanical energy. Initially, the block has kinetic energy due to its speed, and when it compresses the spring and comes to rest, the energy is stored in the potential energy of the compressed spring.
The formula to find the compression distance of a spring is:
Δx = - (m * v^2) / (2 * k)
Where:
Δx is the compression distance of the spring
m is the mass of the block
v is the initial velocity of the block
k is the spring constant
Let's substitute the given values into the formula:
m = 3.7 kg
v = 2.2 m/s
k = 3200 N/m
Plugging these values into the formula:
Δx = - (3.7 * 2.2^2) / (2 * 3200)
Calculating this expression:
Δx = - (3.7 * 4.84) / (2 * 3200)
Δx = -17.948 / 6400
Δx ≈ -0.0028 m
The negative sign indicates that the spring is compressed, which agrees with our understanding that energy is stored in the compressed spring. The compression distance of the spring is approximately 0.0028 meters.