Question

A block of mass 5 kg is placed on a rough inclined surface as shown in the figure.

If \(\vec{F_1}\) is the force required to just move the block up the inclined plane and \(\vec{F_2}\) is the force required to just prevent the block from sliding down, then the value of \(|\vec{F_1}| - |\vec{F_2}|\) is: [Use \(g = 10 \, \text{m/s}^2\)]

• A. \(25 \sqrt{3} \, \text{N}\)
• B. \(5\sqrt{3} \, \text{N}\)
• C. \(\frac{5 \sqrt{3}}{2} \, \text{N}\)
• D. \(10 \, \text{N}\)

Answer 1

The Correct Option is B

To solve this problem, we need to calculate the forces \(|\vec{F_1}|\) and \(|\vec{F_2}|\), and then find the difference \(|\vec{F_1}| - |\vec{F_2}|\).

Given:

• Mass of block (\(m\)) = 5 kg
• Inclination angle (\(\theta\)) = \(30^\circ\)
• Coefficient of friction (\(\mu\)) = 0.1
• Acceleration due to gravity (\(g\)) = 10 m/s²

The forces on the block parallel and perpendicular to the inclined plane are:

\(F_{\text{gravity parallel}} = mg \sin(\theta)\)

\(F_{\text{gravity perpendicular}} = mg \cos(\theta)\)

The limiting friction is given by:

\(F_{\text{friction}} = \mu \cdot F_{\text{gravity perpendicular}} = \mu \cdot mg \cos(\theta)\)

Calculating \(|\vec{F_1}|\) (force to move block up):

To move the block up, apply enough force to overcome both gravity and friction:

\(|\vec{F_1}| = F_{\text{gravity parallel}} + F_{\text{friction}}\)

\(= mg \sin(\theta) + \mu mg \cos(\theta)\)

Substituting values:

\(|\vec{F_1}| = 5 \cdot 10 \cdot \sin(30^\circ) + 0.1 \cdot 5 \cdot 10 \cdot \cos(30^\circ)\)

\(= 50 \cdot 0.5 + 0.1 \cdot 50 \cdot \frac{\sqrt{3}}{2}\)

\(= 25 + 2.5\sqrt{3}\)

Calculating \(|\vec{F_2}|\) (force to prevent sliding down):

To prevent sliding down, counteract gravity by friction:

\(|\vec{F_2}| = F_{\text{gravity parallel}} - F_{\text{friction}}\)

\(= mg \sin(\theta) - \mu mg \cos(\theta)\)

Substituting values:

\(|\vec{F_2}| = 5 \cdot 10 \cdot \sin(30^\circ) - 0.1 \cdot 5 \cdot 10 \cdot \cos(30^\circ)\)

\(= 25 - 2.5\sqrt{3}\)

Calculating the difference:

\(|\vec{F_1}| - |\vec{F_2}| = (25 + 2.5\sqrt{3}) - (25 - 2.5\sqrt{3})\)

\(= 5\sqrt{3} \, \text{N}\)

Thus, the correct answer is \(5\sqrt{3} \, \text{N}\).

Answer 2

The block experiences frictional force due to the roughness of the inclined surface. The kinetic friction force \( f_k \) is given by:

\[ f_k = \mu mg \cos \theta, \]

where \( \mu = 0.1 \) is the coefficient of friction, \( m = 5 \, \text{kg} \), and \( \theta = 30^\circ \).

Calculating \( f_k \):

\[ f_k = 0.1 \times 5 \times 10 \times \cos 30^\circ = 0.1 \times 50 \times \frac{\sqrt{3}}{2} = 2.5 \sqrt{3} \, \text{N}. \]

To move the block up the incline, the force \( F_1 \) must overcome both the component of gravitational force along the incline and the frictional force. Therefore:

\[ F_1 = mg \sin \theta + f_k. \]

Substitute the values:

\[ F_1 = 5 \times 10 \times \sin 30^\circ + 2.5 \sqrt{3} = 25 + 2.5 \sqrt{3} \, \text{N}. \]

To prevent the block from sliding down, the force \( F_2 \) must balance the component of gravitational force along the incline, reduced by the frictional force. Thus:

\[ F_2 = mg \sin \theta - f_k. \]

Substitute the values:

\[ F_2 = 25 - 2.5 \sqrt{3} \, \text{N}. \]

The difference \( |F_1 - F_2| \) is:

\[ |F_1 - F_2| = |(25 + 2.5 \sqrt{3}) - (25 - 2.5 \sqrt{3})| = 5 \sqrt{3} \, \text{N}. \]

Thus, the answer is:

\[ 5 \sqrt{3} \, \text{N}. \]