Problem 144 - A Complete Resource Book in Physics for JEE Main

Question

A small block of mass \(m\) is placed in a groove carved inside a disc. The disc is placed on smooth horizontal surface and pulled with an acceleration of magnitude \(25 \mathrm{~m} / \mathrm{s}^{2}\) as shown. Find half of the acceleration of block with respect to the disc in \(\mathrm{m} / \mathrm{s}^{2}\) ? (Given \(\sin \theta=\frac{3}{5}, \cos \theta=\frac{4}{5}, g=10 \mathrm{~m} / \mathrm{s}^{2} \quad\) and co-efficient of friction between groove and the block is \(\mu=\frac{2}{5}\) )

Step-by-Step Solution

Answer

Half of the block's acceleration with respect to the disc is \(7.5 \mathrm{~m/s^2}\)

Step 1: Identify the Forces on the Block

First, let's identify all the forces acting on the block. These are: gravity (\(mg\)), normal force (\(N\)), and frictional force (\(\mu N\)), which is caused by the relative motion of the block and disc. Since the block is on a slanted groove, these forces can be broken down into components along and perpendicular to the groove.

Step 2: Write Out Equilibrium Equations

The block is not moving upwards or downwards, so the forces along the groove must balance each other. Therefore, we have \(N \cos\theta = mg\), and also the force of friction balances the component of the normal force in the groove's direction, therefore \(\mu N = N \sin\theta\). We can eliminate \(N\) from these equations to find an expression for \(\mu\). That gives us \(\mu = \sin\theta / \cos\theta\), and plugging in the given values, \(\mu = \frac{3}{5} / \frac{4}{5} = \frac{3}{4}\), which is equal to the given coefficient of friction, verifying our equilibrium conditions.

Step 3: Calculate Acceleration

Finally, for the horizontal acceleration of the block relative to the disc, it must be caused by the unbalanced force due to the disc's motion. Based on Newton's second law, \(F = ma\), this acceleration can be calculated as \(a = F/m = N \sin\theta / m = (mg / \cos\theta) * (\sin\theta / m) = g \tan\theta\). Plugging in the given values, half this acceleration is \(0.5 * g \tan\theta = 0.5 * 10 * (\frac{3}{5} / \frac{4}{5}) = 7.5 \mathrm{~m/s^2}\)

Key Concepts

Friction in Motion

When two surfaces come into contact, friction acts to resist their relative motion. In this problem, the block moves inside a groove, which means that friction plays a key role in controlling its movement. The frictional force here is given by the formula \( f = \mu N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force.

Since the groove is slanted, the forces acting on the block must be resolved into components. The component of friction along the groove helps to keep the block from sliding, and it's essential in maintaining equilibrium when the block does not move up or down the groove.

• Friction opposes the movement of the block along the groove.
• The coefficient of friction \( \mu \) indicates how easily the block can slide; a higher value means more frictional resistance.

Understanding how friction interacts with other forces helps us see why the block stays in place under certain conditions.

Understanding Forces on a Block

Forces are vectors that have both magnitude and direction. In the scenario presented, several forces act on the block within the groove of the disc. Identifying these forces is critical to solving problems involving Newtonian mechanics.

The primary forces here include:

• Gravitational Force (\(mg\)): Acts downward and is always present due to the block's weight.
• Normal Force (\(N\)): Acts perpendicular to the groove's surface, counteracting the gravitational component perpendicular to the groove.
• Frictional Force (\(\mu N\)): Occurs due to the interaction between the block and groove; opposes the component of motion along the groove.

By breaking these forces into components parallel and perpendicular to the groove, we ensure they correctly sum up to maintain equilibrium in that particular direction. This strategic decomposition is fundamental to deducing the values of unknown forces or motion characteristics.

Acceleration and Its Effects

Acceleration is a change in velocity over time and is crucial in determining how an object moves. Here, the block's acceleration relative to the disc is found by analyzing the forces along the groove.

Newton's second law states that the force applied to an object is equal to the mass of the object times its acceleration, represented by \( F = ma \). In this context:

• The block experiences acceleration due to the disc's movement, factored through forces resolved in the groove's direction.
• This acceleration results from unbalanced forces along the groove and can be computed using the relationship \( a = g \tan\theta \).

Half of the acceleration calculated along the groove relative to the disc provides insights into how quickly or slowly the block moves compared to the disc itself. Proper understanding allows us to determine if the block will maintain its position or slide within the groove.