a 1.6kg box that is sliding on a friction less surface with the speed of 11m/s approaches a horizontal spring. The spring has a spring constant of 2000N/m. how far will the spring be compressed in stopping the box? How far will the spring be compressed when the box's speed is reduced in half by it initial speed.
To find out how far the spring will be compressed when stopping the box, we can use the principle of conservation of mechanical energy. Initially, the box has kinetic energy due to its motion, but when it comes to a stop, this energy is transformed into potential energy stored in the spring.
To calculate the compression distance, we need to equate the initial kinetic energy of the box with the potential energy stored in the spring. The formula for kinetic energy is given by:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the box, and v is the initial velocity of the box.
Plugging in the values:
m = 1.6 kg
v = 11 m/s
KE = (1/2) * 1.6 kg * (11 m/s)^2
Now, let's calculate the potential energy stored in the spring. The formula for potential energy in a spring is given by:
PE = (1/2) * k * x^2
where PE is the potential energy, k is the spring constant, and x is the compression distance.
Plugging in the values:
k = 2000 N/m (spring constant)
Setting the initial kinetic energy equal to the potential energy:
(1/2) * 1.6 kg * (11 m/s)^2 = (1/2) * 2000 N/m * x^2
Simplifying the equation:
8.8 kg * m^2/s^2 = 1000 N/m * x^2
Dividing both sides by 1000 N/m:
0.0088 kg * m^2/s^2 = x^2
Taking the square root of both sides:
x = √(0.0088 kg * m^2/s^2)
x ≈ 0.094 m (rounded to three decimal places)
Therefore, the spring will be compressed approximately 0.094 meters (or 9.4 centimeters) when stopping the box.
To find out how far the spring will be compressed when the box's speed is reduced to half its initial speed, we can use the principle of conservation of mechanical energy again.
The kinetic energy of the box when its speed is reduced to half is given by:
KE' = (1/2) * m * (v/2)^2
= (1/2) * m * (v^2/4)
= (1/8) * m * v^2
We need to equate this new kinetic energy with the potential energy stored in the spring, as we did before:
(1/8) * 1.6 kg * (11 m/s)^2 = (1/2) * 2000 N/m * x'^2
Simplifying the equation:
(0.125) * 8.8 kg * m^2/s^2 = 1000 N/m * x'^2
1.1 kg * m^2/s^2 = 1000 N/m * x'^2
Dividing both sides by 1000 N/m:
0.0011 kg * m^2/s^2 = x'^2
Taking the square root of both sides:
x' = √(0.0011 kg * m^2/s^2)
x' ≈ 0.033 m (rounded to three decimal places)
Therefore, the spring will be compressed approximately 0.033 meters (or 3.3 centimeters) when the box's speed is reduced to half its initial speed.