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Chapter 7: Problem 20

A body is projected at an angle with the horizontal in the uniform gravitational ficld of thecarth, the angular momentum of the body about the point of projection as it proceds along its path (a) Remains constant (b) Increases (c) Deercases (d) Initially docreases and afier its highost increases

Short Answer

Expert verified
The angular momentum remains constant (option a).

Step by step solution

01

Understanding the Problem

The problem involves a projectile launched at an angle, and we need to determine how its angular momentum changes about the point of projection.
02

Define Angular Momentum

Angular momentum (\( L \)) is defined as the cross product of the radial vector (\( \vec{r} \)) from the point of projection to the body and the linear momentum (\( \vec{p} \)) of the body: \( L = \vec{r} \times \vec{p} \). In a uniform gravitational field, external forces do not exert a torque about the launch point, hence angular momentum remains constant.
03

Analyze Forces and Conditions

In the context of projectile motion, the only force acting on the body is gravity, which acts vertically and passes through the point of projection. Therefore, it doesn't exert a torque about the projection point, confirming that no external torque affects the angular momentum.
04

Conclusion Based on Conservation Laws

Since no external torques are acting on the system, the angular momentum of the body about the point of projection remains constant throughout its motion. This is due to the conservation of angular momentum in the absence of external torques.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion
Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone. Imagine throwing a ball; it follows a curved path called a parabola. This path can be broken down into two main components: horizontal and vertical motion.
  • Horizontal Motion: Once projected, an object moves horizontally at a constant speed because there are no forces acting in this direction (assuming air resistance is negligible).
  • Vertical Motion: The object accelerates downwards due to gravity, which affects its vertical motion, changing the vertical component of velocity over time.
Understanding these components helps us analyze the trajectory and predict where and when the object will land. The angle of projection and initial velocity are key factors influencing its path.
Conservation of Angular Momentum
Angular momentum is a measure of the rotational motion of an object. It's like the rotational equivalent of linear momentum, and it plays a crucial role in systems where rotation is involved.
  • Definition: Angular momentum (\( L \)) is calculated as the cross product of the position vector (\( \vec{r} \)) and linear momentum (\( \vec{p} \)) of the body. Mathematically, it is expressed as \( L = \vec{r} \times \vec{p} \).
  • Conservation Principle: In the absence of external torques, the angular momentum of a system remains constant. This is known as the conservation of angular momentum.
In projectile motion, this means any change in the orientation of the projectile does not come from an outside torque. The angular momentum remains constant unless acted upon by an external force, supporting the fact that the angular momentum of the particle about the launch point remains constant.
Torque
Torque is a measure of the force that can cause an object to rotate about an axis. It's essential for understanding rotational motion.
  • Definition: Torque (\( \tau \)) is given by the product of the force (\( \vec{F} \)) and the lever arm distance (\( \vec{r} \)): \( \tau = \vec{r} \times \vec{F} \). It is the rotational equivalent of force.
  • Effect on Angular Momentum: In systems where no external torque is applied, such as in projectile motion without air resistance, the angular momentum remains unchanged. This absence of external torque is why the projectile's angular momentum about the point of projection remains constant.
Torque can significantly influence mechanical systems, but in the case of a simply thrown object, gravity acts along the line of projection, so no torque is exerted.
Gravitational Field
A gravitational field is a model used to explain the influence of gravity on objects in space. It is essential in understanding projectile motion and the forces acting on the projectile.
  • Uniform Gravitational Field: This assumption implies that the gravitational force is constant in magnitude and direction near the Earth's surface, acting downward.
  • Influence on Projectiles: Within this field, gravity is the sole force acting on a projectile after launch. It does not exert any external torque around the point of projection as its line of action goes through the point.
This uniform gravitational field simplifies calculations and allows us to predict the path of a projectile with great accuracy, trusting that features such as angular momentum remain consistent throughout the motion.

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Most popular questions from this chapter

\(\Lambda\) uniform disc of mass \(m\) and radius \(R\) is rotated about an axis passing through its centre and perpendicular to its plane with an angular velocity \(\omega_{0} .\) It is placed on a rough horizontal plane with the axis of the disc keeping vertical. Coefficient of friction between the disc and the surface is \(\mu\). Find (a) the time when disc stops rotating, (b) the angle rotated by the disc before stopping.

\(\Lambda\) block with a square base measuring \(a \times a\) and height \(h\), is placed on an inclined plane. 'Ihe coefficient of friction is \(\mu\). The angle of inclination \((\theta)\) of the plane is gradually increased. The block will (a) topple before sliding if \(\mu>\frac{a}{h}\) (b) topple before sliding if \(\mu<\frac{a}{h}\) (c) slide before toppling if \(\mu>\frac{a}{h}\) (d) slide before toppling if \(\mu<\frac{a}{h}\)

Statement-1 : \(\Lambda\) ball is rolling on a rough horizontal surface. It gradually slows down and stops. Statement-2 : Force of rolling friction decreases linear velocity.

Two identical cylinders one rotating at angular spoed \(100 \pi \mathrm{rad} / \mathrm{s}\) is bought in contact with other cylinder at rest, sur[ace of both cy linder are rough. If both acquire common acccleration \(2 \pi \mathrm{rad} / \mathrm{s}^{2}\), then (a) Time afler which they achicve cqual angular specd is \(50 \mathrm{~s}\) (b) Time after which they achicve cqual angular specd is \(25 \mathrm{~s}\) (c) Angular spced (common) is \(25 \pi \mathrm{rad} / \mathrm{s}\) (d) Angular spoed (common) is \(50 \pi \mathrm{rad} / \mathrm{s}\)

A unilorm disc kept on a horizontal surfacc is stuck by que at height \(h\) above central linc. Column-I Column-II (a) For no slipping (p) \(h-\frac{R}{2}\) (b) Forward slipping (q) \(h>\frac{R}{2}\) (c) Backward slipping (r) \(h<\frac{R}{2}\) (d) Work done by friction is negative (s) \(h=0\)
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