A block of mass 12.0 kg slides from rest down a frictionless 35.0° incline and is stopped by a strong spring with k = 4.00 104 N/m. The block slides 3.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?
To find out how far the spring has been compressed, we need to use the conservation of mechanical energy.
The initial mechanical energy of the block is equal to the sum of its gravitational potential energy and its kinetic energy. When the block comes to rest against the spring, all of its kinetic energy is converted into potential energy stored in the spring.
The gravitational potential energy of the block can be calculated using the formula:
PE = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the incline.
The kinetic energy of the block can be calculated using the formula:
KE = (1/2) * m * v^2
where v is the velocity of the block.
Since the block starts from rest, its initial velocity is zero. Therefore, the kinetic energy of the block is also zero.
The total mechanical energy of the block is the sum of its initial potential energy and its initial kinetic energy:
E_initial = PE + KE
= m * g * h + 0
= m * g * h
When the block comes to rest against the spring, all of its mechanical energy is converted into potential energy stored in the spring. The potential energy stored in a spring can be calculated using the formula:
PE_spring = (1/2) * k * x^2
where k is the spring constant and x is the displacement (compression or stretching) of the spring.
Since all of the block's mechanical energy is converted to potential energy stored in the spring, we can equate the two:
E_initial = PE_spring
= (1/2) * k * x^2
Rearranging the equation, we can solve for x:
x^2 = (2 * E_initial) / k
x = sqrt((2 * E_initial) / k)
Substituting in the given values, we can calculate x:
x = sqrt((2 * (m * g * h)) / k)
= sqrt((2 * (12.0 kg * 9.8 m/s^2 * 3.00 m)) / (4.00 * 10^4 N/m))
Calculating this expression gives us the distance the spring has been compressed.