A body of mass m=2.6 kg is standing on a ramp inclined in an angle of α=14.9 degrees. The static friction coefficient between the ramp and the body is μ=0.29. The ramp, which is inclined to the left, is in a train, which accelerates to the right. What is the minimal acceleration of the train that will cause the body to slide up the ramp? (use g=10 m/s^2)
To find the minimal acceleration of the train that will cause the body to slide up the ramp, we need to analyze the forces acting on the body.
1. Resolve the gravitational force: Since the body is on an inclined plane, we need to resolve the gravitational force into two components:
- The component perpendicular to the ramp, which is mg * cos(α), where m is the mass of the body and α is the angle of the ramp.
- The component parallel to the ramp, which is mg * sin(α).
2. Determine the limiting frictional force: The maximum static frictional force can be found using the formula: fs = μ * N, where μ is the friction coefficient and N is the normal force. The normal force N is equal to the perpendicular component of the weight, N = mg * cos(α). Therefore, the maximum static frictional force is fs = μ * mg * cos(α).
3. Set up the equations of motion:
- Along the vertical direction, the net force is mg * sin(α) - fs = 0, since the body is not sliding vertically.
- Along the horizontal direction, the net force is the sum of the frictional force and the force due to acceleration, which is fs + ma = m * a. In this case, the acceleration is in the opposite direction of the frictional force to cause the body to slide up.
4. Solve the equations:
Substituting the values into the equations, we have:
mg * sin(α) - fs = 0, and
fs + ma = m * a.
Substituting the values, we get:
mg * sin(α) - μ * mg * cos(α) = 0, and
μ * mg * cos(α) + ma = m * a.
Simplifying, we have:
mg * (sin(α) - μ * cos(α)) = 0, and
a = μ * g * cos(α).
5. Calculate the acceleration:
Substituting the given values, we have:
a = 0.29 * 10 * cos(14.9°).
Evaluating this equation, we find that the minimal acceleration of the train that will cause the body to slide up the ramp is approximately 2.62 m/s^2 (rounded to two decimal places).
To determine the minimal acceleration of the train that will cause the body to slide up the ramp, we need to calculate the force of gravity acting on the body and compare it to the maximum static friction force.
Step 1: Calculate the force of gravity (Fg):
Fg = m * g (where g = 10 m/s^2 is the acceleration due to gravity)
Substituting the given values:
Fg = 2.6 kg * 10 m/s^2
Fg = 26 N
Step 2: Calculate the maximum static friction force (Fs):
Fs = μ * N (where μ = 0.29 is the static friction coefficient and N is the normal force)
To find the normal force, we need to consider the forces acting on the body perpendicular to the ramp:
Normal force (N) = Fg * cos(α) (where α = 14.9 degrees is the angle of inclination)
Substituting the values:
N = 26 N * cos(14.9 degrees)
N = 25.09 N
Now we can calculate the maximum static friction force:
Fs = 0.29 * 25.09 N
Fs ≈ 7.27 N
Step 3: Determine the minimal acceleration of the train (a_min):
The force exerted by the train accelerating to the right is F = m * a, where m = 2.6 kg.
For the body to start sliding up the ramp, the force of gravity acting down the ramp should be greater than the force of static friction. In other words, Fg > Fs.
Therefore, the minimal acceleration (a_min) can be calculated by rearranging the equation:
Fg > Fs
m * a_min > μ * N
a_min > (μ * N) / m
Substituting the given values:
a_min > (0.29 * 25.09 N) / 2.6 kg
a_min > 2.793 / 2.6 m/s^2
a_min > 1.074 m/s^2
Thus, the minimal acceleration of the train that will cause the body to slide up the ramp is approximately 1.074 m/s^2.