A box of mass 2 Kg lies on a rough horizontal floor with the coefficient of friction between the floor and the box being 0.5 (Figure 3). A light string is attached to the box in order to pull the box across the floor. If the tension in the string is TN, find the value that T must exceed for motion to occur if the string is 30º above the horizontal.
To find the minimum tension T that is required for the box to start moving across the floor, we need to consider the forces acting on the box.
First, let's draw a free-body diagram to visualize the forces acting on the box:
- Weight (W) acting vertically downward with a magnitude of 2 Kg * 9.8 m/s^2 = 19.6 N.
- Normal force (N) exerted by the floor on the box, perpendicular to the floor.
- Frictional force (Ff) acting in the opposite direction of motion.
- Tension force (T) in the string, making an angle of 30º with the horizontal.
Now, let's analyze the forces:
- The vertical component of the tension force (Tsinθ = TN) contributes to balancing the weight of the box (W).
- The horizontal component of the tension force (Tcosθ) opposes the frictional force (Ff).
Since the box is on the verge of motion, the frictional force is at its maximum, given by:
Ff = μN,
where μ is the coefficient of friction and N is the normal force.
To find the normal force, we set it equal to the weight of the box, as there is no vertical acceleration:
N = W = 19.6 N.
Substituting this value into the equation for the frictional force, we have:
Ff = 0.5 * N = 0.5 * 19.6 N = 9.8 N.
Since the tension force Tcosθ opposes the frictional force Ff, we can write:
Tcosθ = Ff,
Tcos30º = 9.8 N,
T = 9.8 N / cos30º.
Evaluating this expression, we find:
T ≈ 11.3 N.
Therefore, the tension T must exceed approximately 11.3 N for the box to start moving across the floor.