A 1.50-kg block slides on a rough, horizontal surface. The block hits a spring with a speed of 2.07 m/s and compresses it a distance of 12.2 cm before coming to rest. If the coefficient of kinetic friction between the block and the surface is μk = 0.577, what is the force constant of the spring?
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To find the force constant of the spring, we can follow these steps:
Step 1: Calculate the gravitational force acting on the block.
The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we get:
F_gravity = 1.50 kg * 9.8 m/s^2 = 14.7 N
Step 2: Calculate the normal force acting on the block.
The normal force is the vertical force exerted by the surface on the block, and it is equal in magnitude and opposite in direction to the gravitational force.
Since the block is on a horizontal surface, the normal force is equal to the gravitational force:
F_normal = F_gravity = 14.7 N
Step 3: Calculate the frictional force acting on the block.
The frictional force can be calculated using the formula:
F_friction = μk * F_normal
where μk is the coefficient of kinetic friction.
Plugging in the values, we get:
F_friction = 0.577 * 14.7 N = 8.49 N
Step 4: Calculate the net force acting on the block.
The net force acting on the block can be calculated by subtracting the frictional force from the force applied by the spring:
F_net = F_spring - F_friction
Step 5: Calculate the spring force.
The spring force can be calculated using Hooke's Law, which states:
F_spring = k * x
where k is the force constant of the spring, and x is the distance the spring is compressed.
Plugging in the values, we get:
F_spring = k * 0.122 m
Step 6: Set up the equation for the net force.
Since the block comes to rest, the net force acting on it is zero:
F_net = 0
Step 7: Combine the equations.
Combining the equations from Step 4 and Step 6, we get:
F_spring - F_friction = 0
Step 8: Solve for the force constant.
Substituting F_spring = k * 0.122 m and F_friction = 8.49 N into the equation from Step 7, we get:
k * 0.122 m - 8.49 N = 0
Step 9: Solve for the force constant.
Rearranging the equation from Step 8, we get:
k * 0.122 m = 8.49 N
k = 8.49 N / 0.122 m
Step 10: Calculate the force constant of the spring.
Using a calculator, we can evaluate the expression to find:
k ≈ 69.67 N/m
Therefore, the force constant of the spring is approximately 69.67 N/m.
To find the force constant of the spring, we can use the law of conservation of mechanical energy.
The initial mechanical energy of the block is given by its kinetic energy:
KE_initial = (1/2) * m * v^2
where m is the mass of the block (1.50 kg) and v is the speed of the block (2.07 m/s).
The final mechanical energy of the system is due to the potential energy stored in the compressed spring:
PE_spring = (1/2) * k * x^2
where k is the force constant of the spring (what we want to find) and x is the distance the spring is compressed (12.2 cm = 0.122 m).
The mechanical energy is conserved, so we can set the initial kinetic energy equal to the final potential energy:
(1/2) * m * v^2 = (1/2) * k * x^2
Now, let's calculate the force of friction acting on the block. The force of friction can be determined using the equation:
f_friction = μk * N
where μk is the coefficient of kinetic friction (0.577). The normal force, N, can be calculated using the equation:
N = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting these values into the equation for friction, we have:
f_friction = μk * m * g
Next, let's substitute the expression for the friction force into the equation for the mechanical energy, giving:
(1/2) * m * v^2 = (1/2) * k * x^2 + μk * m * g * x
Now, we can solve this equation to find the force constant, k.