Problem 8 - Master Resource Book in JEE Main Physics

Question

In a children's park, there is a slide which has a total length of \(10 \mathrm{~m}\) and a height of \(8.0 \mathrm{~m}\). A vertical ladder is provided to reach the top. A boy weighing \(200 \mathrm{~N}\) climbs up the ladder to the top of the slide and slides down to the ground. The average friction offered by the slide is three-tenth of his weight. The work done by the slide on the boy as he comes down is \(\begin{array}{lll}\text { (a) zero } & \text { (b) }+600 \mathrm{~J} & \text { (c) }-600 \mathrm{~J}\end{array}\) (d) \(+1600 \mathrm{~J}\)

Step-by-Step Solution

Answer

Option (c) -600 J is correct; the slide does -600 J of work on the boy due to friction.

Step 1: Calculate the force of friction

The problem states that the friction on the slide is three-tenth of the boy's weight. Since the boy's weight is 200 N, the force of friction is given by:\[ F_{friction} = \frac{3}{10} \times 200 = 60 \text{ N} \]

Step 2: Determine the distance along the slide

The boy slides along the length of the slide, which is the hypotenuse of the right triangle formed by the height and base. This length is given directly as 10 m.

Step 3: Calculate the work done by friction

The work done by a force is defined as \( W = F \times d \times \cos \theta \). Since friction acts opposite to the direction of motion, \( \theta = 180^\circ \) and \( \cos 180^\circ = -1 \). Thus, the work done by the frictional force is:\[ W = 60 \times 10 \times (-1) = -600 \text{ J} \]

Step 4: Analyze the options

The calculated work done by friction is equal to -600 J, indicating that the correct option is (c) -600 N. This implies the slide does work against the boy's motion, consuming energy as he slides down.

Key Concepts

Friction Force Calculation

In physics problems involving motion, calculating the friction force is crucial. Friction is a resistive force that opposes the relative motion between two surfaces in contact. It can significantly affect the motion of objects, as seen in this exercise with the slide. Here, the problem states that the friction offered by the slide is three-tenth of the boy's weight. Since the boy weighs 200 N, the friction force can be calculated using a simple multiplication:

• Friction Force, \( F_{friction} = \frac{3}{10} \times 200 = 60 \text{ N} \)

This means a 60 N force is working against the boy as he slides down. Understanding how to calculate friction is vital for solving problems involving motion on rough surfaces. Friction can either be static (preventing motion) or kinetic (acting on moving objects), and knowing the type helps in formulating the right calculations.

Work Done by a Force

Work done by a force in physics is a measure of energy transfer when an object is moved over a distance by an external force. The formula to calculate work done is \( W = F \times d \times \cos \theta \). In this problem, the friction force of 60 N acts over a distance of 10 m along the slide, but opposite the direction of the boy's movement down the slide.

• Angle, \( \theta = 180^\circ \)
• \( \cos 180^\circ = -1 \)

Plugging these components into the formula, we find:

• Work done, \( W = 60 \times 10 \times (-1) = -600 \text{ J} \)

The negative sign indicates that the work is done against the boy's motion, meaning energy is used to overcome this resistive force. Understanding how forces do work helps in gaining insights into energy conservation and transformation in physical systems.

Right Triangle in Physics Problems

In many physics scenarios, particularly those involving inclined planes like slides, right triangles are significant. They help in decomposing vectors and understanding motion along different paths. In this exercise, the slide forms the hypotenuse of a right triangle where the height of the slide is 8 m and the length is 10 m.

• The height represents the vertical component.
• The length of the slide denotes the hypotenuse.

By understanding this triangle, we know the actual path taken by the boy down the slide is 10 m, which is crucial for calculating the work done by friction. Right triangles simplify complex problems by allowing us to apply trigonometric ratios and Pythagorean theorem in calculating forces and distances accurately. They are a fundamental part of solving many mechanics problems in physics.