A 6 kg block slides on a horizontal frictionless surface with a speed of 1.5 m/s . It is brought momentarily to rest when it hits a bumper that compresses a spring. The acceleration of gravity is 9.8 m/s2 .How much is the spring compressed if its
force constant is 7 kN/m?
Answer in units of m.
My answers are incorrect please help!
1/2 mv^2= 1/2 kx^2
1ST ANSWER (1/2)(6)(1.5^2)=(1/2)(7)X^2
X=1.3887 WRONG
2ND ANSWER (1/2)(6)(1.5^2)=(1/2)(.007)X^2 X=43.9155 WRONG
K=7kN/m = 7000N/m
Otherwise,ok.
Thank you!
To solve this problem, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the block will be equal to the potential energy stored in the compressed spring.
The initial kinetic energy is given by:
KE = 1/2 * mass * velocity^2
Substituting the given values:
KE = 1/2 * 6 kg * (1.5 m/s)^2
KE = 6.75 J
The potential energy stored in the spring can be calculated using Hooke's Law:
PE = 1/2 * k * x^2
We are given the force constant, k, as 7 kN/m, which needs to be converted to N/m:
k = 7 kN/m * 1000 N/1 kN
k = 7000 N/m
Substituting the values:
PE = 1/2 * 7000 N/m * x^2
Now, equating the initial kinetic energy with the potential energy, we have:
6.75 J = 1/2 * 7000 N/m * x^2
Simplifying the equation:
6.75 J = 3500 N/m * x^2
Dividing both sides by 3500 N/m:
0.00192857 J/m = x^2
Taking the square root of both sides to solve for x:
x = √0.00192857 J/m
x ≈ 0.0439 m
Therefore, the spring is compressed by approximately 0.0439 m.
To find the correct answer, let's break down the problem step by step.
Firstly, we know that the block is momentarily brought to rest, which means that its initial kinetic energy is converted into potential energy stored in the compressed spring, as well as potential energy due to gravity.
The formula for gravitational potential energy is given by:
PE_gravity = mgh
where m is the mass (6 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height. In this case, the height is zero since the block is on a horizontal surface. Therefore, the gravitational potential energy is zero.
Next, let's consider the potential energy stored in the compressed spring. This can be calculated using the formula:
PE_spring = (1/2)kx^2
where k is the force constant of the spring (7 kN/m), and x is the displacement or compression of the spring.
The initial kinetic energy of the block, which is equal to the potential energy stored in the spring, can be expressed as:
KE_initial = PE_spring = (1/2)kx^2
Now, let's substitute the given values into the equation:
(1/2)mv^2 = (1/2)kx^2
Rearranging the equation, we can solve for x:
(1/2)6(1.5^2) = (1/2)(7x10^3)x^2
Simplifying the equation:
13.5 = 3500x^2
Dividing both sides by 3500:
0.003857 = x^2
Taking the square root of both sides:
x ≈ 0.062 m (or x ≈ -0.062 m, but we can take the positive value as the compression of the spring cannot be negative)
Therefore, the spring is compressed approximately 0.062 m.