A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$. A brick $P$ of mass $m$ is placed on the plane. The coefficient of friction between $P$ and the plane is $\mu$. Brick $P$ is in equilibrium and on the point of sliding down the plane. Brick $P$ is modelled as a particle. Using the model, (a) Find, in terms of $m$ and $g$, the magnitude of the normal reaction of the plane on brick $P$. (b) show that $\mu = \frac{3}{4}$. For parts (c) and (d), you are not required to do any further calculations. Brick $P$ is now removed from the plane and a much heavier brick $Q$ is placed on the plane. The coefficient of friction between $Q$ and the plane is also $\frac{3}{4}$. (c) Explain briefly why brick $Q$ will remain at rest on the plane. Brick $Q$ is now projected with speed $0.5$ m s$^{-1}$ down a line of greatest slope of the plane. Brick $Q$ is modelled as a particle. (d) describe the motion of brick $Q$, giving a reason for your answer.

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A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$ - Edexcel - A-Level Maths Mechanics - Question 1 - 2020 - Paper 1

Question 1

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A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$. A brick $P$ of mass $m$ is placed on the plane. The coeffici... show full transcript

Worked Solution & Example Answer:A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$ - Edexcel - A-Level Maths Mechanics - Question 1 - 2020 - Paper 1

Step 1

(a) Find, in terms of m and g, the magnitude of the normal reaction of the plane on brick P.

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Answer

To find the normal reaction $R$ on brick $P$ , we resolve the forces perpendicular to the inclined plane. The weight of the brick $P$ can be expressed as:

$mg$

The component of the weight acting perpendicular to the plane is given by:

$mg \cos \alpha$

Thus, the normal reaction can be defined as:

$R = mg \cos \alpha$

Substituting $\tan \alpha = \frac{3}{4}$ leads to:

Calculate $\alpha$ using:

$\alpha = \tan^{-1}\left(\frac{3}{4}\right)$

Hence, we can find:

$R = mg \left(\frac{4}{5}\right)$
(This is derived from the right triangle where the hypotenuse will be $5$ and the adjacent side will be $4$ .)

Step 2

(b) show that μ = 3/4.

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Answer

To determine the coefficient of friction $\mu$ , we resolve the forces acting parallel to the plane. The force acting along the inclined plane is:

$F = mg \sin \alpha$

Using the friction force:

$F_{f} = \mu R$

Setting $F$ equal to the frictional force gives:

$mg \sin \alpha = \mu R$

Since we already established that:

$R = mg \cos \alpha$

We can substitute this into our friction equation, leading to:

$mg \sin \alpha = \mu (mg \cos \alpha)$

Dividing both sides by $mg$ (assuming $m \neq 0$ ), we simplify it to:

$\sin \alpha = \mu \cos \alpha$

Thus:

$\mu = \frac{\sin \alpha}{\cos \alpha} = \tan \alpha$

Since $\tan \alpha = \frac{3}{4}$ , we arrive at:

$\mu = \frac{3}{4}$

Step 3

(c) Explain briefly why brick Q will remain at rest on the plane.

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Answer

Brick $Q$ will remain at rest on the plane due to the balance of forces acting upon it. The force of friction acting on $Q$ will equal the component of its weight acting down the slope. As long as the weight component and the force of friction are equal, it will not slide.

Step 4

(d) describe the motion of brick Q, giving a reason for your answer.

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Answer

Brick $Q$ will slide down the plane with a constant speed. This occurs because the force of gravity acting down the plane is balanced by the frictional force acting up the plane, resulting in no net force and hence no acceleration.

A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$ - Edexcel - A-Level Maths Mechanics - Question 1 - 2020 - Paper 1

Worked Solution & Example Answer:A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$ - Edexcel - A-Level Maths Mechanics - Question 1 - 2020 - Paper 1

(a) Find, in terms of m and g, the magnitude of the normal reaction of the plane on brick P.

(b) show that μ = 3/4.

(c) Explain briefly why brick Q will remain at rest on the plane.

(d) describe the motion of brick Q, giving a reason for your answer.

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