Question

A block of mass \(4 \mathrm{~kg}\) is placed on a rough horizontal plane. A time dependent force \(\mathrm{F}=\mathrm{Kt}^{2}\) acts on a block, where \(\mathrm{k}=2 \mathrm{~N} / \mathrm{s}^{2}\), co-efficient of friction \(\mu=0.8\). Force of friction between the block and the plane at $\mathrm{t}=2 \mathrm{~S}$ is.... (A) \(32 \mathrm{~N}\) (B) \(4 \mathrm{~N}\) (C) \(2 \mathrm{~N}\) (D) \(8 \mathrm{~N}\)

Short answer

The force of friction between the block and the plane at \(t=2~\mathrm{s}\) is \(8~\mathrm{N}\). The correct answer is (D) \(8~\mathrm{N}\).

Step-by-step solution

Write down the time-dependent force equation

The time-dependent force is given by \(\mathrm{F}=\mathrm{Kt}^2\), where \(\mathrm{K} = 2~\mathrm{N}/\mathrm{s}^2\). We need to find the force of friction acting on the block at \(t = 2~\mathrm{s}\).

Calculate the force at the given time

To find the force acting on the block at \(t = 2~\mathrm{s}\), we'll substitute the given time into the time-dependent force equation: \(\mathrm{F} = 2 \times (2~\mathrm{s})^2\) \(\mathrm{F} = 2 \times 4~\mathrm{N}\) \(\mathrm{F} = 8~\mathrm{N}\) So the force acting on the block at \(t = 2~\mathrm{s}\) is \(8~\mathrm{N}\).

Find the normal force

Since the block is on a horizontal plane, there is no vertical acceleration or force. Therefore, the normal force (\(N\)) equals the weight of the block: \(N = mg\) \(N = (4~\mathrm{kg})(9.81~\mathrm{m/s^2})\) \(N = 39.24~\mathrm{N}\)

Calculate the maximum force of static friction

Using the coefficient of friction \(\mu = 0.8\), we can find the maximum force of static friction (\(F_{s_{max}}\)) between the block and the plane: \(F_{s_{max}} = \mu N\) \(F_{s_{max}} = (0.8)(39.24~\mathrm{N})\) \(F_{s_{max}} = 31.392~\mathrm{N}\)

Compare the force with the force of static friction

We can now compare the force acting on the block at \(t = 2~\mathrm{s}\) (\(8~\mathrm{N}\)) with the maximum force of static friction (\(31.392~\mathrm{N}\)). Since the force is less than the maximum force of static friction, the block doesn't move and the force of friction is equal to the force acting on the block. Therefore, the force of friction at \(t = 2~\mathrm{s}\) is \(8~\mathrm{N}\). The correct answer is (D) \(8~\mathrm{N}\).