Two blocks, each with a mass = 0.168 kg, can slide without friction on a horizontal surface. Initially, block 1 is in motion with a speed = 1.28 m/s; block 2 is at rest. When block 1 collides with block 2, a spring bumper on block 1 is compressed. Maximum compression of the spring occurs when the two blocks move with the same speed, /2 = 0.640 m/s. If the maximum compression of the spring is 1.55 cm, what is its force constant?
At maximum compression, the spring potential energy will equal the drop in Total kinetic energy. Total KE goes from (M/2)Vo^2 to 2*(M/2)*(Vo/2)^2
= (M/4)Vo^2/4
M is the mass of one block; Vo is its initial speed of block 1
(M/2)Vo^2/4 = (k/2)X^2
Solve for k
Im still confused. Ive tried solving for k a couple of different ways and the website tells me that I am wrong. My answers have ranged from .0286 to 791.1 N/m. Ive tried converting 1.55 cm to m, and it still says that I am wrong.
X = 1.55*10^-2 m
M = 0.168 kg
Vo = 1.28 m/s
k = (Vo/X)^2*M = 1146 N/m
You apparently did not use the formula I provided.
To find the force constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement.
The formula for the force exerted by a spring is given by F = -kx, where F is the force, k is the force constant, and x is the displacement.
In this case, the maximum compression of the spring is given as 1.55 cm, which we need to convert to meters to maintain consistent units. Therefore, 1.55 cm = 0.0155 m.
First, let's find the change in velocity of block 1 during the collision. This can be done using the principle of conservation of momentum.
The initial momentum of block 1 is given by:
p1_initial = m1 * v1_initial = (0.168 kg) * (1.28 m/s)
Since block 2 is initially at rest, its initial momentum is zero:
p2_initial = m2 * v2_initial = 0
The total initial momentum is therefore:
p_initial = p1_initial + p2_initial
After the collision, the final momentum of both blocks is equal and given by:
p_final = (0.168 kg + 0.168 kg) * (0.64 m/s) = 0.336 kg * 0.64 m/s
According to the law of conservation of momentum, the total momentum before and after the collision must be the same, i.e.:
p_initial = p_final
Using this equation, we can solve for the initial momentum of block 1 (p1_initial) and substitute it into the formula for kinetic energy (KE = 0.5 * m * v^2).
Now that we have the initial and final kinetic energies, the change in kinetic energy (ΔKE) can be calculated as:
ΔKE = KE_final - KE_initial
During the collision, the compression of the spring is fully converted into potential energy (PE) stored in the spring. Therefore, the change in potential energy of the spring is equal to the change in kinetic energy of the blocks.
Since the maximum compression of the spring is reached when they have the same speed, both blocks will have the same final kinetic energy, which means ΔKE = 0.
Now we can write the equation:
0 = KE_final - KE_initial
Rearranging the equation:
KE_initial = KE_final
Solving for KE_initial to find the initial kinetic energy of block 1:
KE_initial = 0.5 * m1 * v1_initial^2
Now we have all the necessary information to calculate the force constant of the spring.
Since the change in potential energy is equal to the change in kinetic energy, we can use the formula for potential energy stored in a spring:
PE = 0.5 * k * x^2
Substituting the values we know:
0 = 0.5 * k * x^2
Solving for k to find the force constant of the spring:
k = 0 / (0.5 * x^2)
Substituting the maximum compression value, x = 0.0155 m, into the equation:
k = 0 / (0.5 * (0.0155 m)^2)
Simplifying the equation:
k = 0
Therefore, the force constant of the spring is 0.