Question and Problem information is in "Description" below.

Question

A spring with a spring constant of k = 191 N/m is initially compressed by a block at a distance d = 0.39 m from its unstretched length. The block is on a horizontal surface with coefficients of static and kinetic friction µ_s and µ_k and has a mass of m = 1 kg.

Assuming the block has just begun to move and the coefficient of kinetic friction is µ_k = 0.2, what is the block's acceleration in meters per second squared?

Answer 1

To find the block's acceleration when it has just begun to move, we need to consider the forces acting on the block. These forces include the force of the spring, the force of gravity, and the force of kinetic friction.

1. The force of the spring (Fs) can be calculated using Hooke's law:

Fs = -k * d

Where:

- k = Spring constant = 191 N/m

- d = Compression of the spring = 0.39 m

Fs = -191 N/m * 0.39 m = -74.49 N (negative because it acts in the opposite direction of compression)

2. The force of gravity (Fg) acting on the block is given by:

Fg = m * g

Where:

- m = Mass of the block = 1 kg

- g = Acceleration due to gravity = 9.8 m/s²

Fg = 1 kg * 9.8 m/s² = 9.8 N

3. The force of kinetic friction (Fk) is given by:

Fk = µk * Fn

Where:

- µk = Coefficient of kinetic friction = 0.2

- Fn = Normal force (which is equal to the weight of the block since it's on a horizontal surface)

Fn = Fg = 9.8 N

Now, calculate Fk:

Fk = 0.2 * 9.8 N = 1.96 N

Now, we can use Newton's second law to find the acceleration (a) of the block:

ΣF = ma

ΣF is the net force, which is the sum of the spring force and the force of kinetic friction, taking directions into account:

ΣF = Fs + Fk

ΣF = (-74.49 N) + (1.96 N) = -72.53 N (negative because it acts in the direction of compression)

Now, use Newton's second law:

ma = -72.53 N

a = (-72.53 N) / (1 kg) ≈ -72.53 m/s²

The block's acceleration is approximately -72.53 m/s². Since the negative sign indicates that the acceleration is in the direction opposite to the initial compression of the spring, this means the block is moving to the left.

Answer 2

m = 1 kg, spring constant k = 191 N/m, spring compression x = –0.39 m

We're also given the coefficient of kinetic friction μ_k = 0.2

Regarding + and – signs: I'm calling the direction of motion (to the right) positive.

Horizontal Forces

F_sp = –kx =

f_k = –μ_kmg =

Net Force

F_net = F_sp + f_k = –kx – μ_kmg

Acceleration

a = F_net /m = –kx/m – μ_kg

Plug in values to calculate acceleration.