A 11.1-kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of 3.12 m/s in 1.17 s. In the process, the spring is stretched by 0.173 m. The block is then pulled at a constant speed of 3.12 m/s, during which time the spring is stretched by only 0.0507 m. Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table.
To solve this problem, we can break it down into two parts: finding the spring constant and then finding the coefficient of kinetic friction.
(a) To find the spring constant:
We can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
The equation for Hooke's Law is: F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
Step 1: Calculate the force exerted by the spring.
Given: mass of the block (m) = 11.1 kg, acceleration (a) = v/t = 3.12 m/s / 1.17 s = 2.67 m/s^2
Using Newton's second law (F = ma), we can calculate the force exerted by the block.
F_block = m * a = 11.1 kg * 2.67 m/s^2 = 29.637 N
Step 2: Calculate the force exerted by the stretched spring.
Given: displacement (x) = 0.173 m
Using Hooke's Law (F = -kx), we can calculate the force exerted by the spring.
F_spring = -k * x
Step 3: Equate the forces exerted by the block and the spring.
Since the block and the spring are both exerting forces in opposite directions, we have:
F_block = -F_spring
29.637 N = -(-k * 0.173 m)
k = (29.637 N) / (0.173 m)
k ≈ 171 N/m
So, the spring constant is approximately 171 N/m.
(b) To find the coefficient of kinetic friction:
To find the coefficient of kinetic friction, we need to analyze the block while it is moving at a constant speed.
Step 1: Calculate the force exerted by the stretched spring.
Given: displacement (x) = 0.0507 m
Using Hooke's Law (F = -kx), we can calculate the force exerted by the spring.
F_spring = -k * x = -171 N/m * 0.0507 m
Step 2: Determine the total force acting on the block.
Since the block is moving at a constant speed, the net force acting on it must be zero.
The total force acting on the block is the sum of the force exerted by the spring and the force of friction.
0 = F_spring + F_friction
Step 3: Solve for the force of friction.
F_friction = -F_spring
F_friction = -(-171 N/m * 0.0507 m)
Step 4: Use the force of friction to find the normal force.
The force of friction can be calculated using the equation F_friction = μ * F_normal, where μ is the coefficient of kinetic friction and F_normal is the normal force acting on the block.
Since the block is on a horizontal table, the normal force is equal to the weight of the block.
F_normal = m * g = 11.1 kg * 9.8 m/s^2
Step 5: Solve for the coefficient of kinetic friction.
F_friction = μ * F_normal
-(-171 N/m * 0.0507 m) = μ * (11.1 kg * 9.8 m/s^2)
Now, we can solve the equation to find μ, the coefficient of kinetic friction.
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