A spring that has a force constant of 1100 N/m is mounted vertically on the ground. A block of mass 1.90 kg is dropped from rest from height of 1.65 m above the free end of the spring. By what distance does the spring compress?
To calculate the distance by which the spring compresses, we can use the principle of conservation of mechanical energy.
Step 1: Calculate the potential energy of the block when it starts to fall.
The potential energy (PE) of an object is given by the formula PE = m * g * h, where m is the mass (1.90 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (1.65 m).
PE = 1.90 kg * 9.8 m/s^2 * 1.65 m
PE = 30.586 J
Step 2: Calculate the potential energy of the compressed spring.
The potential energy (PE) stored in a spring is given by the formula PE = (1/2) * k * x^2, where k is the force constant of the spring (1100 N/m), and x is the distance by which the spring compresses.
PE = (1/2) * 1100 N/m * x^2
Step 3: Apply conservation of mechanical energy to find x.
Since the mechanical energy is conserved, we can equate the potential energy of the falling block to the potential energy of the compressed spring.
PE (block) = PE (spring)
30.586 J = (1/2) * 1100 N/m * x^2
Step 4: Solve for x.
Rearrange the equation to solve for x.
x^2 = (2 * 30.586 J) / (1100 N/m)
x^2 = 0.0557 m^2
Take the square root of both sides to find the distance by which the spring compresses.
x = √(0.0557 m^2)
x ≈ 0.236 m
Therefore, the spring compresses by approximately 0.236 meters.
To find the distance by which the spring compresses, we can use the conservation of mechanical energy.
The mechanical energy of the system is conserved, so the initial potential energy of the block when it is at height H above the free end of the spring is equal to the sum of the final potential energy of the block when it compresses the spring and the final elastic potential energy stored in the spring.
The potential energy of the block at height H is given by the equation:
PE_initial = m * g * h
= 1.90 kg * 9.8 m/s^2 * 1.65 m
= 31.819 J
When the block compresses the spring, it gains potential energy and the spring gains elastic potential energy. The potential energy gained by the block is converted into elastic potential energy of the spring.
The potential energy gained by the block is given by:
PE_block_gained = m * g * x
where x is the distance by which the spring compresses.
The elastic potential energy of the spring is given by:
PE_spring = (1/2) * k * x^2
where k is the force constant of the spring.
Since the mechanical energy is conserved, we can set up the equation:
PE_initial = PE_block_gained + PE_spring
Plugging in the values we know:
31.819 J = 1.90 kg * 9.8 m/s^2 * x + (1/2) * 1100 N/m * x^2
Rearranging the equation:
0 = 1100 N/m * x^2 + 1.90 kg * 9.8 m/s^2 * x - 31.819 J
This is a quadratic equation in terms of x. We can solve it by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 1100 N/m, b = 1.90 kg * 9.8 m/s^2, and c = -31.819 J.
Plugging in the values:
x = (-(1.90 kg * 9.8 m/s^2) ± √((1.90 kg * 9.8 m/s^2)^2 - 4 * 1100 N/m * (-31.819 J))) / (2 * 1100 N/m)
Calculating the value inside the square root:
√((1.90 kg * 9.8 m/s^2)^2 - 4 * 1100 N/m * (-31.819 J)) = √(72.4184) = 8.5144
Substituting this value back in:
x = (-(1.90 kg * 9.8 m/s^2) ± 8.5144) / (2 * 1100 N/m)
Now we can calculate the two possible values of x:
x1 ≈ (-18.62 + 8.5144) / 2200 ≈ -0.004 m ≈ -4 mm (ignoring the negative sign)
x2 ≈ (-18.62 - 8.5144) / 2200 ≈ -0.015 m ≈ -15 mm (ignoring the negative sign)
Since the negative value doesn't make sense in this context, we take the positive value:
Therefore, the spring compresses by approximately 4 mm (or 0.004 m).