An 18.0 kg box is released on a 35.0° incline and accelerates down the incline at 0.269 m/s2. Find the friction force impeding its motion.
How large is the coefficient of friction?
To find the friction force impeding the motion of the box, we first need to calculate the gravitational force component pulling the box down the incline.
1. Calculate the gravitational force pulling the box down the incline:
The gravitational force (Fg) can be calculated using the formula:
Fg = m * g
where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s²).
In this case, m = 18.0 kg.
Fg = 18.0 kg * 9.8 m/s²
Fg = 176.4 N
2. Calculate the component of gravitational force acting parallel to the incline:
Since the incline is at an angle of 35.0°, only a portion of the gravitational force will act parallel to the incline.
The component of gravitational force acting parallel to the incline (Fg_parallel) can be calculated using the formula:
Fg_parallel = Fg * sin(theta)
where theta is the angle of the incline.
In this case, theta = 35.0°.
Fg_parallel = 176.4 N * sin(35.0°)
Fg_parallel = 100.4 N (rounding to one decimal place)
3. Calculate the net force parallel to the incline:
The net force parallel to the incline (F_net) can be calculated using the formula:
F_net = m * a
where m is the mass of the box and a is the acceleration down the incline.
In this case, m = 18.0 kg and a = 0.269 m/s².
F_net = 18.0 kg * 0.269 m/s²
F_net = 4.8 N (rounding to one decimal place)
4. Calculate the friction force impeding the motion:
Since the box is accelerating down the incline, there must be a friction force acting in the opposite direction.
The friction force (F_friction) can be calculated using the formula:
F_friction = Fg_parallel - F_net
F_friction = 100.4 N - 4.8 N
F_friction = 95.6 N (rounding to one decimal place)
Therefore, the friction force impeding the motion of the box is 95.6 N.
To find the coefficient of friction, we can use the equation:
F_friction = μ * N
where μ is the coefficient of friction and N is the normal force (perpendicular to the incline).
Since the box is on an incline, the normal force can be calculated using:
N = Fg * cos(theta)
where theta is the angle of the incline.
In this case, theta = 35.0°.
N = 176.4 N * cos(35.0°)
N = 144.1 N (rounding to one decimal place)
Substituting the values into the equation F_friction = μ * N, we get:
95.6 N = μ * 144.1 N
Solving for μ:
μ = 95.6 N / 144.1 N
μ ≈ 0.664 (rounding to three decimal places)
Therefore, the coefficient of friction is approximately 0.664.
Netforcedownplane= mass*acceleration
mg*SinTheta-mu*mgCosTheta= mass*a
solve for mu.