Physics Mechanics Quiz

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Questions and Answers

What is the dimensional formula for the Van der Waals constant 'a'?

The dimensional formula for 'a' is $M L^5 T^{-2}$.

Explain the difference between average velocity and average speed.

Average velocity is the ratio of total displacement to total time, while average speed is the ratio of total path length to total time.

How is instantaneous velocity defined?

Instantaneous velocity is the velocity of an object at a specific instant in time, reflecting its speed at that moment.

What condition characterizes uniform motion?

<p>Uniform motion occurs when an object covers equal distances in equal time intervals.</p> Signup and view all the answers

State the principle of homogeneity regarding dimensions in equations.

<p>The principle of homogeneity states that quantities in an equation must have the same dimensions to be added or subtracted.</p> Signup and view all the answers

How is average speed calculated?

<p>Average speed is calculated by dividing the total path length by the total time interval.</p> Signup and view all the answers

In the context of the Van der Waals equation, what role do constants 'a' and 'b' play?

<p>'a' corrects for intermolecular forces, while 'b' accounts for the volume occupied by gas molecules.</p> Signup and view all the answers

What is the impulse imparted to the ball when a batsman hits it back with a speed of 12 m/s?

<p>3.6 N s</p> Signup and view all the answers

State Newton's third law of motion and its significance.

<p>For every action, there is an equal and opposite reaction.</p> Signup and view all the answers

Explain why action and reaction forces do not cancel each other out.

<p>They act on different bodies, so they do not cancel each other despite being equal and opposite.</p> Signup and view all the answers

What is the weight reading on a scale for a man of mass 70 kg in a lift moving upwards at a uniform speed of 10 m/s?

<p>The scale would read 70 kg.</p> Signup and view all the answers

What would happen to the scale reading if the lift falls freely under gravity?

<p>The scale would read zero.</p> Signup and view all the answers

What is the formula for the time of flight T of a projectile in terms of its initial velocity and launch angle?

<p>The formula is $T = \frac{2u \sin \theta}{g}$.</p> Signup and view all the answers

How is the horizontal range R of a projectile calculated?

<p>The horizontal range is calculated as $R = u \cos \theta \times T$ where $T$ is the time of flight.</p> Signup and view all the answers

For which angle is the range of a projectile maximized when launched with a given velocity?

<p>The range R is maximized when the launch angle is $\theta = 45^\circ$.</p> Signup and view all the answers

Show that the ranges for angles $\theta$ and $(90 - \theta)$ are equal.

<p>For angles $\theta$ and $(90 - \theta)$, $R = \frac{u^2 \sin 2\theta}{g}$ and $R = \frac{u^2 \sin (180 - 2\theta)}{g}$ yield the same result since $\sin(180 - 2\theta) = \sin 2\theta$.</p> Signup and view all the answers

What is defined as the maximum height H reached by a projectile?

<p>The maximum height H is defined as the peak vertical distance attained by the projectile during its flight.</p> Signup and view all the answers

In the context of projectile motion, what role does gravity play during the trajectory?

<p>Gravity acts as a downward acceleration that influences the vertical motion of the projectile, affecting both time of flight and maximum height.</p> Signup and view all the answers

What effect does increasing the initial launch speed have on the time of flight T?

<p>Increasing the initial launch speed u directly increases the time of flight T, as seen in the formula $T = \frac{2u \sin \theta}{g}$.</p> Signup and view all the answers

How does the angle of projection affect the horizontal range and maximum height of a projectile?

<p>The angle affects the trade-off between horizontal range and maximum height, with angles around $45^\circ$ optimizing both.</p> Signup and view all the answers

What property of vector addition states that A + B = B + A?

<p>The commutative property of vector addition.</p> Signup and view all the answers

How can the difference of two vectors A and B be defined?

<p>A - B can be defined as A + (-B).</p> Signup and view all the answers

What does the parallelogram law of vector addition state?

<p>It states that the resultant of two vectors is represented by the diagonal of a parallelogram formed by the vectors.</p> Signup and view all the answers

What is a unit vector and how is it represented mathematically?

<p>A unit vector is a vector of unit magnitude that specifies direction, often represented as Ā = |𝐴̅|𝐴̂.</p> Signup and view all the answers

Which symbols represent the unit vectors along the x-, y-, and z-axes?

<p>The unit vectors are denoted by î, ĵ, and k̂ respectively.</p> Signup and view all the answers

How can a vector A be resolved into components in a rectangular coordinate system?

<p>Vector A can be resolved into components along the unit vectors î and ĵ.</p> Signup and view all the answers

What equation represents the magnitude of the resultant vector R when adding vectors A and B?

<p>The magnitude is given by R = √(A² + B² + 2AB cos θ).</p> Signup and view all the answers

What condition must unit vectors satisfy regarding their magnitudes?

<p>Unit vectors must satisfy |î| = |ĵ| = |k̂| = 1.</p> Signup and view all the answers

What does the equation 𝑙=𝑟×𝑝 represent in physics?

<p>It represents the relationship between angular momentum (𝑙), the position vector (𝑟), and linear momentum (𝑝).</p> Signup and view all the answers

What is the significance of differentiating the angular momentum equation?

<p>Differentiating provides the rate of change of angular momentum with respect to time, which relates to torque.</p> Signup and view all the answers

Explain the role of torque in the context of angular momentum.

<p>Torque affects the rate of change of angular momentum, as shown by the relation d𝑙/d𝑡 = 𝜏.</p> Signup and view all the answers

How is torque mathematically expressed in terms of other variables?

<p>Torque can be expressed as τ = r × F, where r is the position vector and F is the applied force.</p> Signup and view all the answers

What does the term 'p = mv' imply in the context of angular momentum?

<p>It indicates that linear momentum (p) is the product of mass (m) and velocity (v) of a particle.</p> Signup and view all the answers

In the absence of torque, what happens to angular momentum over time?

<p>Angular momentum remains constant if no external torque is acting on the system.</p> Signup and view all the answers

What does r x p represent in the differentiation of angular momentum?

<p>It represents the angular momentum due to linear momentum at a specific position vector.</p> Signup and view all the answers

Why is the equation d𝑙/d𝑡 = 0 important in rotational dynamics?

<p>It indicates that in the absence of torque, the angular momentum does not change over time.</p> Signup and view all the answers

What is the relationship between torque and angular momentum in a system of particles?

<p>Torque is the rate of change of angular momentum, expressed as $\vec{\tau} = \frac{d\vec{L}}{dt}$.</p> Signup and view all the answers

Explain the law of conservation of angular momentum.

<p>If the total external torque on a system of particles is zero, then the total angular momentum remains constant.</p> Signup and view all the answers

How do you calculate the torque of a force acting on a particle?

<p>Torque can be calculated using the cross product: $\vec{\tau} = \vec{r} \times \vec{F}$.</p> Signup and view all the answers

What conditions must be satisfied for a rigid body to be in mechanical equilibrium?

<p>A rigid body is in mechanical equilibrium if both the total external force and total external torque acting on it are zero.</p> Signup and view all the answers

Define translational equilibrium in the context of rigid bodies.

<p>Translational equilibrium occurs when the total external force on a rigid body is zero, indicating no change in linear momentum.</p> Signup and view all the answers

What is rotational equilibrium and how is it achieved?

<p>Rotational equilibrium is achieved when the total external torque on the body is zero, causing no change in angular momentum.</p> Signup and view all the answers

Can a body exist in partial equilibrium? Provide an example.

<p>Yes, a body can be in partial equilibrium, such as when it is in translational equilibrium but not in rotational equilibrium.</p> Signup and view all the answers

Using the given vectors, calculate the torque $ au$ of the force acting on the particle at position vector $\vec{r}$.

<p>The torque is $\tau = 2\hat{i} + 12\hat{j} + 10\hat{k}$.</p> Signup and view all the answers

Flashcards

Average Velocity

The ratio of the total displacement to the total time interval.

Average Speed

The ratio of the total path length (distance traveled) to the total time interval.

Uniform Motion

Motion where an object covers equal distances in equal intervals of time.

Instantaneous Velocity

The velocity of an object at a specific instant in time.

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Time of Flight (T)

The time it takes for a projectile to travel from its launch point to the point where it hits the ground.

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Horizontal Range (R)

The horizontal distance traveled by a projectile during its flight.

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Maximum Height (H)

The maximum vertical distance a projectile reaches during its flight.

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Range at Complementary Angles

The horizontal distance traveled by a projectile is the same when launched at angles θ and (90-θ).

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Time of Flight Formula

The formula to calculate the time of flight of a projectile.

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Horizontal Range Formula

The formula to calculate the horizontal range of a projectile.

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Maximum Height Formula

The formula to calculate the maximum height of a projectile.

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Maximum Range Condition

The maximum horizontal range is achieved when the launch angle is 45 degrees.

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Impulse

The change in momentum of an object caused by a force acting on it over a period of time.

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Newton's Third Law of Motion

For every action, there is an equal and opposite reaction.

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Weight of a body

The normal force exerted by a surface on an object in contact with it.

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Weight of a body in a lift

The weight of a body appears to be different when the lift is accelerating, either upwards or downwards.

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Action-Reaction Forces

The force on body A by body B is equal and opposite to the force on body B by body A.

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Commutative Property of Vector Addition

Vector addition is commutative, meaning the order in which you add vectors doesn't affect the result. It's like saying 2 + 3 is the same as 3 + 2.

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Associative Property of Vector Addition

Vector addition also follows the associative law, meaning you can group vectors in any order when adding them. It's like saying (2 + 3) + 4 is the same as 2 + (3 + 4).

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Vector Subtraction

Subtracting vectors is the same as adding the negative of the second vector. It's like saying A - B is the same as A + (-B).

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Parallelogram Law of Vector Addition

A vector can be represented by a diagonal line in a parallelogram where the adjacent sides represent the vectors to be added. This diagonal line represents their resultant, both in magnitude and direction.

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Unit Vector

A vector with a magnitude of 1, used to indicate direction. Unit vectors have no units or dimensions.

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Vector Representation using Unit Vectors

Any vector can be expressed as the product of its magnitude and the unit vector pointing in its direction. Think of a vector's magnitude as its 'strength' and the unit vector as its 'direction'.

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Resolution of a Vector

Breaking down a vector into its component vectors along orthogonal unit vectors (usually î, ĵ, and k̂ for the x, y, and z axes).

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Vector Addition – Analytical Method

A method to add vectors analytically, using their components and the angle between them. This involves resolving the vectors into their components and then using trigonometry to find the magnitude and direction of the resultant.

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Angular momentum (l)

The measure of an object's tendency to rotate, calculated as the product of its moment of inertia and its angular velocity.

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Torque (τ)

A force that causes an object to rotate around an axis.

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Time rate of change of angular momentum (dl/dt)

The rate of change of angular momentum over time.

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Relationship between Torque and Angular Momentum

The mathematical relationship between the time rate of change of angular momentum and the torque acting on an object.

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Rotational analogue of Newton's 2nd Law

The rotational analogue (equivalent) of Newton's second law, relating torque to the rate of change of angular momentum.

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Angular momentum equation (l=r x p)

The vector product of the object's position vector (r) and its linear momentum (p).

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Perpendicular component of linear momentum

The component of the linear momentum that is perpendicular to the position vector.

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Angle (θ) between position and momentum

The angle between the position vector and the linear momentum vector.

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Torque

The rate of change of angular momentum with respect to time. It is a vector quantity and its direction is perpendicular to both the force and the position vector.

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Angular Momentum

The product of the moment of inertia and the angular velocity of a system of particles. It is a vector quantity and its direction is along the axis of rotation.

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Law of Conservation of Angular Momentum

The angular momentum of a system remains constant if the net external torque acting on the system is zero. In simple terms, if no external forces are trying to rotate the system, its rotation will stay the same.

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Mechanical Equilibrium

A rigid body is in mechanical equilibrium if it is in both translational and rotational equilibrium. This means the linear and angular momentum of the body are constant over time.

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Translational Equilibrium

A rigid body is in translational equilibrium when the net external force acting on it is zero. This means the linear velocity of the body is constant over time, regardless of the body's rotation.

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Rotational Equilibrium

A rigid body is in rotational equilibrium when the net external torque acting on it is zero. This means the angular velocity of the body is constant over time, regardless of the body's linear motion.

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Partial Equilibrium

A body can be in either translational or rotational equilibrium, but not both. For instance, a spinning object can have zero net force acting on it, but still be rotating.

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Torque of a Force

The product of the force and the perpendicular distance from the axis of rotation to the line of action of the force. It's a vector quantity, and its direction is given by the right-hand rule.

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Study Notes

General Study Notes

  • Study actively, engaging with the materials. Focus on understanding concepts, not just memorizing facts.
  • Create a dedicated study space free from distractions.
  • Take breaks to avoid burnout and maintain focus.
  • Review material regularly, ideally daily, to reinforce learning.
  • Seek clarification when needed.
  • Practice problem-solving to apply concepts.
  • Use various study aids (flashcards, summaries, diagrams) to enhance understanding.
  • Connect new information to prior knowledge for stronger retention.
  • Collaborate with classmates for mutual learning and support.

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