a 10.0 kg box is placed on a ramp that is inclined at 30.0 degrees to the horizontal.
Determine the force friction required to keep the block stationary.
Determine the minimum coefficient of friction required to keep the box stationary.
the force component of the box's weight along the ramp is
... m g sin(30º)
the frictional force is
... μ m g cos(30º)
To determine the force of friction required to keep the box stationary on the inclined ramp, we need to consider the forces acting on the box. There are two main forces to consider: the force of gravity acting vertically downward and the force of friction acting horizontally along the ramp.
First, let's find the force of gravity acting on the box. The force of gravity can be calculated using the equation:
F_gravity = m * g
where m is the mass of the box (10.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = 10.0 kg * 9.8 m/s^2
F_gravity = 98 N
Next, let's determine the force of friction required to keep the box stationary. The force of friction is given by the equation:
F_friction = μ * F_normal
where μ is the coefficient of friction and F_normal is the normal force, which is the component of the force of gravity perpendicular to the inclined ramp. The normal force can be calculated using the equation:
F_normal = F_gravity * cos(θ)
where θ is the angle of the incline (30.0 degrees).
F_normal = 98 N * cos(30.0 degrees)
F_normal = 98 N * 0.866
F_normal ≈ 84.9 N
Now, we can find the force of friction:
F_friction = μ * F_normal
To keep the box stationary, the force of friction must be equal and opposite to the force component pulling the box down the incline. This component can be calculated using:
F_component = F_gravity * sin(θ)
F_component = 98 N * sin(30.0 degrees)
F_component = 98 N * 0.5
F_component = 49 N
Since the box is stationary, the force of friction must be equal to the force component:
F_friction = 49 N
Finally, let's determine the minimum coefficient of friction required to keep the box stationary. We rearrange the equation for force of friction:
μ = F_friction / F_normal
μ = 49 N / 84.9 N
μ ≈ 0.577
Therefore, the minimum coefficient of friction required to keep the box stationary on the inclined ramp is approximately 0.577.