Final answer:
To calculate the time before the box moves without slipping on the conveyor belt, the force of friction and the box's acceleration due to this force are considered. However, because the mass of the box is not given, we can only specify the method to find the time (using motion equations) but cannot provide a definitive answer from the options given.
Step-by-step explanation:
To find out how long it will take before the box moves without slipping, we can use the concept of friction and the equations of motion. The force of friction (f) that acts on the box and opposes its motion is given by the frictional force equation:
f = μ * N
where μ is the coefficient of friction between the box and the conveyor belt, and N is the normal force. In this case, the normal force is equal to the weight of the box, since it is lying on a horizontal surface. As the conveyor belt starts moving, the frictional force will cause the box to accelerate until its speed matches that of the conveyor belt (3.4 m/s).
The acceleration (A) can be calculated using:
A = f / m
Where m is the mass of the box. Since acceleration is constant, we can use one of the equations of motion to calculate the time (t) it takes for the box to reach the speed of the conveyor belt:
v = u + A * t
Where v is the final velocity, u is the initial velocity (which is 0 since the box is initially at rest), and t is the time. Rearranging the above formula to solve for time, we have:
t = (v - u) / A
Once we have the acceleration, we can substitute it into this formula, with v = 3.4 m/s and u = 0 m/s, to find the value of t.
To find the mass of the box and the normal force, we may need additional information, which is not provided. However, the mass of the box will cancel out in the equation for acceleration since it's both in the numerator of the force of friction and in the denominator of the acceleration formula. Hence, the time to reach the non-slipping speed will not depend on the mass of the box.
Since we don't have the mass of the box, we cannot calculate the exact time. Therefore, we are unable to determine which answer option is correct.