Problem 7 - Master Resource Book in JEE Main Physics

Question

A block of mass \(10 \mathrm{~kg}\) slides down a rough slope which is inclined at \(45^{\circ}\) to the horizontal. The coefficient of sliding friction is \(0.30\). When the bloek has slide \(5 \mathrm{~m}\), the work done on the block by the force of friction is nearly (a) \(115 \mathrm{~J}\) (b) \(75 \sqrt{2} \mathrm{~J}\) (c) \(321.4 \mathrm{~J}\) (d) \(-321.4 \mathrm{~J}\)

Step-by-Step Solution

Answer

The work done by friction is approximately \(-104\, \text{J}\), closest to none of the provided options.

Step 1: Identify the Forces

The block is moving down a slope, so the forces acting on it are gravity, normal force, and friction. Friction acts against the direction of motion.

Step 2: Calculate the Normal Force

The normal force can be found using the component of the gravitational force perpendicular to the incline. The normal force is given by:\[ F_n = mg \cos \theta \]where \(m = 10\, \text{kg}, g = 9.8\, \text{m/s}^2, \theta = 45^\circ\). Substituting the values gives:\[ F_n = 10 \cdot 9.8 \cdot \cos(45^\circ) \approx 10 \cdot 9.8 \cdot 0.707 \approx 69.3\, \text{N} \]

Step 3: Calculate the Force of Friction

The force of friction \( F_f \) can be calculated using the coefficient of friction \( \mu \).\[ F_f = \mu F_n = 0.30 \times 69.3 \approx 20.8\, \text{N} \]

Step 4: Calculate the Work Done by Friction

Work is calculated as \( W = Fd \cos \phi \) where \( F \) is the force of friction, \( d \) is the distance, and \( \phi \) is the angle between the force and displacement vectors (which is \(180^\circ\), because friction opposes the motion). Thus,\[ W = 20.8 \times 5 \times \cos(180^\circ) = 20.8 \times 5 \times (-1) = -104\, \text{J} \]

Step 5: Match the Calculated Work to Given Options

The calculated work done by friction is \(-104\, \text{J}\), which does not exactly match any of the given options. However, since the options must be considered carefully, ensure calculations were done accurately or revisit the multiple-choice answers for potential rounding.

Key Concepts

Inclined Plane Problems

An inclined plane is a flat surface that is tilted at an angle, other than 90 degrees, compared to a horizontal plane. When dealing with problems involving inclined planes, it is crucial to analyze the forces acting on an object that is placed on or moving along the inclined plane. This analysis often involves:

• Gravity - causing a component of the object's weight to pull it down the slope.
• Normal Force - which is perpendicular to the surface of the incline and supports the object against gravity.
• Friction - opposing the motion of the object down the incline.

These forces need to be accounted for to determine various outcomes such as the acceleration of the object, the work done by the forces, or even the final velocity of the object after traveling a certain distance. Understanding inclined plane problems often provides a foundation for solving more complex physics problems that involve motion and forces.

Friction and Its Calculation

Friction is the resistive force that acts against the relative motion of two surfaces in contact. In the context of an inclined plane, friction works to oppose the sliding motion of an object down the slope. The strength of this frictional force depends on the following factors:

• The normal force, which exerted between the two surfaces.
• The coefficient of friction (\(\mu\), which quantifies how rough or smooth the surfaces are.

Friction can be calculated using the formula:\[ F_f = \mu F_n \]where \( F_n \) is the normal force, and \( \mu \) is the coefficient of friction. The frictional force is an essential factor in calculating the net force acting on the object and, consequently, the work done by this force when the object moves a specific distance along the incline.

Normal Force

The normal force is a crucial concept when dealing with problems involving inclined planes. It is the force exerted by a surface, perpendicular to itself, to support the weight of an object placed upon it. On a flat horizontal surface, the normal force is equal and opposite to the gravitational force acting on the object. However, on an inclined plane, the normal force is only a component of the object's weight. It can be calculated using the formula:\[ F_n = mg \cos \theta \]where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of the incline. Calculating the normal force accurately is essential because it affects the frictional force acting on the object, influencing how easily the object slides down the plane and the overall work done by friction.