To calculate the time it takes for the cylinder to start rolling without slipping, we can use the concept of torque and Newton's second law of motion for rotational motion.

First, let's calculate the moment of inertia of the cylinder. The moment of inertia of a solid cylinder about its central axis can be given by the formula:

I = (1/2) * m * r^2

Where:
m = mass of the cylinder
r = radius of the cylinder

Substituting the given values, we get:

I = (1/2) * 10 kg * (0.5 m)^2
I = 1.25 kg.m^2

Next, we need to calculate the net torque acting on the cylinder. The net torque is given by the product of force (F) and the radius (r) at which the force is applied. In this case, the force is applied at the center of the cylinder, so the torque is:

τ = F * r
τ = 100 N * 0.5 m
τ = 50 N.m

Finally, we can use Newton's second law for rotational motion, which states that the net torque acting on a body is equal to its moment of inertia multiplied by its angular acceleration (α). Here, we can assume that when the cylinder starts rolling without slipping, the angular acceleration is zero.

τ = I * α
α = τ / I
α = 50 N.m / 1.25 kg.m^2
α = 40 rad/s^2

Now, we can calculate the time it takes for the cylinder to start rolling without slipping. The angular acceleration is equal to the change in angular velocity divided by the time taken to achieve this change. Since the initial angular velocity is zero, we have:

α = Δω / Δt
0 = ω / Δt

Solving for Δt:

Δt = ω / 0
Δt = undefined

The time it takes for the cylinder to start rolling without slipping is undefined because the angular acceleration is zero. This means that the cylinder will not start rolling when the constant horizontal force of 100 N is applied.

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