Problem 16 - Applied Mathematics

Question

A smooth plane is fixed at an inclination \(30^{\circ}\) with its lower edge at a height \(a\) above a horizontal table. Two particles \(\mathrm{P}\) and \(\mathrm{Q}\), each of mass \(m\), are connected by a light inextensible string of length \(2 a\), and \(\mathrm{P}\) is held at the lower edge of the inclined plane while \(\mathbf{Q}\) rests on the table vertically below \(\mathrm{P}\). The particle \(\mathrm{P}\) is then projected with velocity \(u(u>\sqrt{g a}\) ) upwards along a line of greatest slope of the plane. Find the impulsive tension in the string when \(\mathrm{Q}\) is jerked into motion. Determine the magnitude of \(u\) if \(\mathrm{Q}\) just reaches the lower edge of the plane, and the tension in the string while \(Q\) is moving.

Step-by-Step Solution

Answer

Impulsive tension, I=ma/2, magnitude u=8ga/3. The tension in the string while Q is moving is 2P (2 masses).

Step 1: Understand the setup

Visualize the plane inclined at an angle \(30^{\circ}\) with the lower edge at height \(a\) above a horizontal table. Two particles, P and Q, each of mass m are connected by a string of length \(2a\). Particle P starts at the edge of the incline and is projected upwards, and Q is initially below P on the table.

Step 2: Constraints and motion

Particle P moves up the plane until it has moved a distance causing the string to lift Q off the table. At this point, the length of the string allows Q to rest below P on the table before it is jerked into motion. The incline angle is \(30^{\circ}\), so the components of the forces and motion need considering.

Step 3: Conditions to jerk Q into motion

Determine the height P must reach along the inclined plane for Q to move: \[ h = a \tan(30^{\circ}) = a \frac{1}{\sqrt{3}}. \]

Step 4: Calculate the distance moved by P

Use trigonometry to find the distance (\(L\)) P travels up the plane to reach the required height: \[ L = h \times \frac{1}{\tan(30^{\text{degree}})}= a \]

Step 5: Kinematics of P's motion

Use the equations of motion to find the velocity of P when Q is jerked into motion: \[ v^2 = u^2 - 2a g \frac{L}{m} \Rightarrow v^2 = u^2 - 2a g a \]

Step 6: Find Impulsive Tension

When P jerks Q into motion, use impulse-momentum relations to find tension T: \[ I = \frac{m(u^2 - v^2)}{2a} \]

Step 7: Determine magnitude of u when Q reaches the edge

Determine u using the condition provided: \[ u = \frac{2g L}{3}\]

Step 8: Tension while Q is moving

While Q is moving, the force in the string needs to be calculated: \[ T = \frac{{mg}}{2} \]

Key Concepts

Inclined Plane

In physics, an inclined plane is a flat surface tilted at an angle, referred to as the angle of inclination. Here, the plane is inclined at an angle of \(30^{\circ}\). Because it's smooth, friction is negligible. This makes calculations easier by focusing only on gravitational and normal forces. The key details are:

• A smooth, frictionless surface.
• An angle of \(30^{\circ}\) with the horizontal.
• Particles P and Q connected by a string.

Understanding the geometry of the inclined plane is crucial. When projecting particle P upwards, consider the incline to determine force components acting parallel and perpendicular to the plane. The gravitational force component pulling P downwards along the slope is \(mg \sin(30^{\circ})\). This component influences P's motion up the plane.

Kinematics

Kinematics deals with motion without considering its causes. In this problem, we're concerned with particle P's motion up the inclined plane. Let's break it down further:

• Initial velocity \(u\) of particle P.
• Distance traveled along the plane before the string affects particle Q.
• Relationships between speed, distance, and height.

The projection of P means its initial kinetic energy must overcome both its initial potential energy difference and the work done against gravity. The equation of motion, used here, is:\[\ v^2 = u^2 - 2a g a.\]Here, \(v\) is the velocity of P at the moment Q is jerked into motion. The distance travelled by P, expressed as \(L = a= \frac{1}{\tan(30^{\circ})}\), confirms that P covers a vertical height of \(a / \sqrt{3}\) before picking up Q.

Impulsive Forces

Impulsive forces are large forces acting over short time periods, causing a change in momentum. To find the impulsive tension when Q starts moving, consider the scenario just before and just after Q is pulled. The string

Trigonometry in Physics

Trigonometry helps in understanding relationships between angles and distances in physics problems involving inclined planes. Key principles used here include:

• \tan(30^{\circ}) = \frac{1}{\sqrt{3}}.
• \sin(30^{\circ}) = \frac{1}{2}.
• Pythagorean identities for relating height, length, and angles.

In our scenario, the plane's inclination and distance calculations use basic trigonometric ratios. For example, determining the height: \(h = a \tan(30^{\circ}) = \frac{a}{\sqrt{3}}\), shows how trigonometry simplifies finding elevations and corresponding lengths. These principles solve spatial relationships necessary for understanding particle pathways on inclines.