A 15.0-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 5.00 m/s in 0.460 s. In the process, the spring is stretched by 0.250 m. The block is then pulled at a constant speed of 5.00 m/s, during which time the spring is stretched by only 0.0500 m.
(a) Find the spring constant of the spring.
Answer in N/m
(b) Find the coefficient of kinetic friction between the block and the table.
To find the spring constant of the spring, we can use the equation for the acceleration of the block:
a = F_net / m
Where:
a = acceleration of the block
F_net = net force acting on the block
m = mass of the block
In this case, the net force acting on the block is the force exerted by the spring, given by Hooke's Law:
F_spring = -k * x
Where:
F_spring = force exerted by the spring
k = spring constant
x = displacement of the spring from its equilibrium position
Given that the block reaches a speed of 5.00 m/s in 0.460 s, we can calculate the acceleration using the equation:
a = Δv / Δt
Where:
Δv = change in velocity
Δt = change in time
In this case, Δv = 5.00 m/s and Δt = 0.460 s. Plugging these values into the equation, we find:
a = 5.00 m/s / 0.460 s = 10.87 m/s²
Now, we can set the net force equal to the force exerted by the spring and solve for the spring constant:
F_net = F_spring
m * a = -k * x
Plugging in the given values, we have:
15.0 kg * 10.87 m/s² = -k * 0.250 m
Simplifying the equation, we find:
k = -(15.0 kg * 10.87 m/s²) / 0.250 m
Calculating this expression, we get:
k ≈ -630 N/m
Note that the negative sign indicates that the direction of the force exerted by the spring is opposite to the displacement.
For part (b), we can find the coefficient of kinetic friction between the block and the table using the following equation:
F_friction = μ_k * F_normal
Where:
F_friction = force of friction
μ_k = coefficient of kinetic friction
F_normal = normal force
In this case, since the block is pulled at a constant speed of 5.00 m/s, its acceleration is zero. Therefore, the net force acting on the block is zero. This means that the force of friction is equal in magnitude but opposite in direction to the force exerted by the spring:
F_friction = F_spring
Using the equation for the force exerted by the spring, we can write:
μ_k * F_normal = -k * x
Since the block is not accelerating, the normal force and the force of gravity are equal in magnitude:
F_normal = mg
Where:
m = mass of the block
g = acceleration due to gravity
Plugging in the given values, we have:
μ_k * mg = -k * x
Using the previously calculated value of the spring constant, k ≈ -630 N/m, and the given value of the displacement of the spring, x = 0.0500 m, we can solve for the coefficient of kinetic friction:
μ_k * (15.0 kg * 9.8 m/s²) = -(-630 N/m) * 0.0500 m
Simplifying the equation, we find:
μ_k ≈ (-(-630 N/m) * 0.0500 m) / (15.0 kg * 9.8 m/s²)
Calculating this expression, we get:
μ_k ≈ 0.020
Therefore, the coefficient of kinetic friction between the block and the table is approximately 0.020.
To solve this problem, we can start by finding the acceleration of the block when it reaches a speed of 5.00 m/s in 0.460 s. We can use the formula:
v = u + at
where:
v = final velocity = 5.00 m/s
u = initial velocity = 0 m/s
t = time = 0.460 s
Substituting the given values, we can solve for acceleration (a):
5.00 = 0 + a * 0.460
a = 5.00 / 0.460
a = 10.87 m/s^2
Now, let's find the spring constant (k). The spring constant can be found using Hooke's Law:
F = -kx
where:
F = force applied to the spring = ma (from Newton's second law)
x = displacement of the spring = 0.250 m
m = mass of the block = 15.0 kg
Substituting the values, we get:
ma = -kx
15.0 * 10.87 = -k * 0.250
k = -15.0 * 10.87 / 0.250
k = -652.25 N/m (Note: The negative sign indicates that the force applied by the spring is in the opposite direction of the displacement)
Therefore, the spring constant of the spring is 652.25 N/m.
Moving on to part (b), we can find the coefficient of kinetic friction using the formula:
F_friction = μ * N
where:
F_friction = force of friction
μ = coefficient of kinetic friction (what we want to find)
N = normal force = mg
Since the block is moving at a constant speed of 5.00 m/s, the acceleration is zero. Therefore, the net force acting on the block in the horizontal direction is zero. This means the force applied by the spring is equal to the force of friction:
kx = F_friction
where:
k = spring constant = 652.25 N/m (we found this in part (a))
x = displacement of the spring = 0.0500 m
Substituting the values, we get:
652.25 * 0.0500 = μ * mg
To find μ, we need to find the value of g (acceleration due to gravity) and substitute the known values:
g = 9.8 m/s^2
m = 15.0 kg
0.0500 * 652.25 = μ * 15.0 * 9.8
μ = (0.0500 * 652.25) / (15.0 * 9.8)
Using a calculator, we can calculate μ:
μ ≈ 0.231
Therefore, the coefficient of kinetic friction between the block and the table is approximately 0.231.