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 and determine when the force of static friction is overcome.
1. First, let's draw a free body diagram of the body on the ramp:
- There is the weight of the body acting vertically downward, mg = 2.6 kg * 10 m/s^2 = 26 N.
- There is the normal force perpendicular to the ramp, which is equal in magnitude but opposite in direction to the vertical component of the weight. Normal force = mg * cos(α).
- There is the force of static friction, which opposes the motion and acts up the ramp. Static friction force = μ * (mg * cos(α)).
2. The net force acting down the ramp is given by the difference between the gravitational force component down the ramp and the static friction force:
Net force = mg * sin(α) - μ * (mg * cos(α)).
3. For the body to slide up the ramp, the net force acting up the ramp should be greater than zero:
Net force > 0.
4. Substituting the values into the equation, we have:
mg * sin(α) - μ * (mg * cos(α)) > 0.
5. Simplifying the equation:
2.6 kg * 10 m/s^2 * sin(14.9°) - 0.29 * (2.6 kg * 10 m/s^2 * cos(14.9°)) > 0.
6. Calculate the value:
4.357 N - 6.666 N > 0.
7. Since the net force acting up the ramp is negative, it means that the static friction force is greater than the gravitational force component down the ramp. This indicates that the body is not able to slide up the ramp.
Therefore, there is no minimal acceleration of the train that will cause the body to slide up the ramp. Instead, the body will remain stationary or slide down the ramp depending on the acceleration of the train.