Question

A block is on a horizontal slab which is moving horizontally with a simple harmonic motion of frequency two oscillations per second . The cofficient of static friction between the block and the slab is 0.50. How great the amplitude can be if the block does not slip along the slab ?

Answer 1

3.1cm

Answer 2

mw^2A=<umg.

A=3.1cm

Answer 3

[AI-generated answer — Explain Bot]

To determine the maximum amplitude of the simple harmonic motion for which the block does not slip along the slab, we need to consider the maximum value of frictional force that can be exerted on the block without it moving.

The maximum static frictional force (𝑓𝑠𝑡𝑎𝑡) is given by:

𝑓𝑠𝑡𝑎𝑡 = 𝜇𝑠 × 𝑁

Where:
- 𝜇𝑠 is the coefficient of static friction (given as 0.50)
- 𝑁 is the normal force exerted on the block by the slab

In this case, the normal force is equal to the weight of the block (𝑊 = 𝑚 × 𝑔), where 𝑚 is the mass of the block and 𝑔 is the acceleration due to gravity.

Now, we know that the maximum acceleration experienced by an object in simple harmonic motion is given by:

𝑎𝑚𝑎𝑥 = 𝑤 × 𝐴

Where:
- 𝑎𝑚𝑎𝑥 is the maximum acceleration
- 𝑤 is the angular frequency of the motion
- 𝐴 is the amplitude of the motion

In this case, the angular frequency (𝑤) is given by 2𝜋𝑓, where 𝑓 is the frequency of 2 oscillations per second.

Now, the maximum acceleration can also be expressed as 𝑤² × 𝐴.

For the block to remain in contact with the slab without slipping, the force of friction (𝑓) must be equal to or greater than the maximum static frictional force (𝑓𝑠𝑡𝑎𝑡):

𝑓 ≥ 𝑓𝑠𝑡𝑎𝑡

Now, the force of friction can be expressed as 𝑓 = 𝑚 × 𝑎, where 𝑎 is the maximum acceleration.

Therefore:

𝑚 × 𝑎 ≥ 𝜇𝑠 × 𝑁

Substituting 𝑚 × 𝑎 = 𝑤² × 𝑚 × 𝐴, and 𝑁 = 𝑚 × 𝑔:

𝑤² × 𝑚 × 𝐴 ≥ 𝜇𝑠 × 𝑚 × 𝑔

Simplifying the equation by canceling out the mass (𝑚):

𝑤² × 𝐴 ≥ 𝜇𝑠 × 𝑔

Finally, solving for the amplitude (𝐴):

𝐴 ≥ (𝜇𝑠 × 𝑔) / 𝑤²

Plugging in the given values of 𝜇𝑠 = 0.50, 𝑔 ≈ 9.8 m/s², and 𝑤 = 2𝜋𝑓 (where 𝑓 = 2 oscillations per second), you can calculate the minimum value of the amplitude (𝐴) for which the block does not slip along the slab.

Answer 4

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