Kinematics and Kinetics MCQ Quiz - Objective Question with Answer for Kinematics and Kinetics - Download Free PDF

Explanation:

The Rate of Decay of Oscillations

• The rate of decay of oscillations refers to the decrease in the amplitude of oscillatory motion over time due to energy dissipation. In mechanical systems, this decay is primarily caused by damping forces that convert the mechanical energy of oscillation into other forms, such as heat. The term used to quantify this rate of decay is known as the "Logarithmic Decrement."

Logarithmic Decrement:

• Logarithmic decrement is a measure that quantifies the rate at which the amplitude of an oscillatory system decreases between successive cycles. It is commonly represented by the Greek letter δ (delta). This parameter is particularly useful in analyzing underdamped systems, where oscillations persist but gradually reduce in amplitude. Logarithmic decrement is mathematically expressed as:

Formula:

Logarithmic Decrement (δ) = ln(A₁/A₂)

Where:

• A₁ = Amplitude of the first oscillation
• A₂ = Amplitude of the second oscillation
• ln = Natural logarithm

Working Principle: Logarithmic decrement is derived from the damping characteristics of the system. When a system is set into oscillatory motion, damping forces—such as viscous damping—cause the amplitude of the oscillation to decrease with each successive cycle. The logarithmic decrement provides a quantitative measure of this decay and helps engineers and analysts determine the damping efficiency of the system.

Applications:

• Used in vibration analysis to evaluate the damping characteristics of structures and machinery.
• Helpful in designing mechanical systems to ensure stability and mitigate excessive vibrations.
• Applied in various engineering fields such as civil engineering, automotive engineering, and aerospace engineering to optimize system performance.

Concept:

We analyze the motion of a particle with variable acceleration to determine its velocity when the acceleration becomes zero.

Given:

• Acceleration function:
• Initial conditions: at

Step 1: Find the position when acceleration is zero

Set

Step 2: Relate acceleration to velocity and position

Using the chain rule, acceleration can be expressed as:

Separate variables and integrate:

Step 3: Integrate both sides

Concept:

When a body moves along a curved path with constant tangential acceleration, the total acceleration has two components:

1. Tangential acceleration due to change in speed.

2. Normal (centripetal) acceleration due to change in direction, given by:

Calculation: 

Given:

Final speed after 60 seconds: v = 18 km/h = 5 m/s

Radius of curve: r = 200 m

Time = 60 s, Initial speed = 0 (starts from rest)

Step 1: Calculate tangential acceleration

Step 2: Speed at 30 seconds:

Step 3: Normal acceleration

Concept:

To find the time of meeting of the two shots, we consider the relative motion of the projectiles in the horizontal direction.

Given:

Distance between guns, d = 40 m

Velocity of shot from Gun A, v_A = 350 m/s

Velocity of shot from Gun B, v_B = 300 m/s

Angle of projection, 

Calculation:

Resolving velocities into horizontal components:

m/s

m/s

The relative velocity in the horizontal direction is:

m/s

Using the formula,

Approximating  ,

 seconds

Concept:

When two bodies collide and move together after the collision, the principle of conservation of momentum is applied.

The equation for conservation of momentum is given by:

Where:

• kg (mass of first sphere)
• m/s (velocity of first sphere before collision)
• kg (mass of second sphere)
• m/s (velocity of second sphere before collision, since it is at rest)
• is the common velocity after collision

Calculation:

Applying the conservation of momentum equation:

m/s

Concept:

• Average velocity = total displacement/ total time duration
• Equation of motion:
• v = u + at
• v² = u² + 2as
• s = ut + 1/2 at2

Calculation:

Given:

Time interval = 5 s & 1 s, Initial velocity u = 0, and, Acceleration a = 2 m/s²

​When an object starts from rest, then the total distance covered in time 1 sec and 5 sec is,

s = ut + 1/2 at2

Object is at rest, so, u = 0 m/s.

s₂ - s₁ = 24 m

Total time taken, t = t₂ - t₁ = 5 - 1 = 4 sec

Average velocity = total displacement/ total time duration

Average velocity = 24/4 = 6 m/s

Average velocity between time 1 s and 5 s = 6 m/s

Concept:

Rate of change of velocity is known as acceleration. Its unit is m/s2. It is a vector quantity.

a = change in velocity/time

Equations of motion:

• v = u + at
• v2 – u2 = 2as

Calculation:

Given:

u = 36 km/h = 10 m/s; S = 125 m ; v = 54 km/h = 15 m/s,  t = ?

v2 – u2 = 2as

v = u + at

CONCEPT:

Centripetal Acceleration (a_c): 

• Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.
• It always acts on the object along the radius towards the center of the circular path.
• The magnitude of centripetal acceleration,

Where v = velocity of the object and r = radius

Tangential acceleration (a_t):

• It acts along the tangent to the circular path in the plane of the circular path.
• Mathematically Tangential acceleration is written as

Where α = angular acceleration and r = radius

CALCULATION:

Given – v = 10 m/s, r = 25 m and a_t = 3 m/s²

• Net acceleration is the resultant acceleration of centripetal acceleration and tangential acceleration i.e.,

Centripetal Acceleration (a_c):

Hence, net acceleration 

Concept:

Work done = Area under force deflection diagram

Calculation:

Given:

Gradual loading

Force = 100 N, deflection = 100 mm = 0.1 m

Explanation:

When the Force F is suddenly removed, then due to W, the rod is in rotating condition with angular acceleration 

Thus the equation of motion:

Also, the centre of the rod accelerates with linear acceleration a;

Explanation:

Non-conservative forces:

• A non-conservative force is one for which work depends on the path taken.
• Friction is an example of a non-conservative force that changes mechanical energy into thermal energy.
• Work Wnc done by a non-conservative force changes the mechanical energy of a system.
• In equation form, Wnc = ΔKE + ΔPE or, equivalently, KEi + PEi = Wnc + KEf + PEf.

Conservative forces:

• A conservative force is one, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. 
• Example potential energy, gravitational force etc.
• Work done by conservative force is equal to decrease in potential energy.

Concept:

A body moving in a horizontal circle has only kinetic energy stored in it.

Centripetal force = 

where m is mass, v is the velocity of particle, and r is the radius.

The kinetic energy of a body is given by:

Calculation:

Given:

Centripetal force = 

Centripetal force = 

Don't Confuse with the negative sign in the value of K.E as constant C must be negative here so that K.E become positive. 

Concept:

Absolute velocity is given by:

v_x = velocity component in x direction = dx/dt

v_y = velocity component in y direction = dy/dt

Calculation:

At t = 6 sec

x = 4 – 9t

v_x = dx/dt = -9

y = t²

v_y = dy/dt = 2t = 2(6) = 12

Explanation:

Acceleration:

• The object under motion can undergo a change in its speed. The measure of the rate of change in its speed along with direction with respect to time is called acceleration.
• The motion of the object can be linear or circular.

Linear acceleration:

• The acceleration involved in linear motion is called linear acceleration.
• Here the acceleration is only due to the change in speed and no acceleration due to change in direction, therefore no radial component of acceleration.

Circular acceleration:

• The acceleration involved in a circular motion is called angular acceleration.
• In a circular motion, the acceleration experienced by the body towards the centre is called the centripetal acceleration which can be resolved into two-component.
• A radial component and a tangential component depending upon the type of motion.

Radial acceleration (a_r):

• The acceleration of the object along the radius, directed towards the centre is called radial acceleration.

Tangential acceleration (a_t):

• The tangential component is defined as the component of angular acceleration tangential to the circular path.

Concept:

Projectile motion: 

When a particle is projected obliquely near the earth's surface, it moves simultaneously in horizontal and vertical directions. The path of such a particle is called projectile and the motion is called projectile motion.

Range of projectile:

• The horizontal range of a projectile is the distance along the horizontal plane it would travel, before reaching the same vertical position as it started from.

Formulae in projectile motion:

where u = projected speed, θ = angle at which an object is thrown from the ground and g = acceleration due to gravity = 9.8 m/s².

Calculation:

Given:

Range of a Projectile motion is given by (R):

For horizontal distance to be maximum:

sin 2θ = 1

∴ sin 2θ = sin 90° 

∴ θ = 45°.

∴ to take the longest possible jump, an athlete should make an angle of 45° with the ground.