Force and Motion for Engineering Physics

The principle of conservation of linear momentum states that if no external net force acts on a closed (isolated) system of particles, the total linear momentum of the system remains constant in time.

The impulse of a force is the product of the force and the time interval during which it acts. It measures how much the force changes an object’s momentum. Impulse is a vector and its SI unit is N·s (same as kg·m/s).

A gun recoils after firing due to the principle of conservation of linear momentum.

Before firing, the gun and the bullet are both at rest, so the total momentum of the system is zero.

When the bullet is fired, it is pushed forward with a certain momentum. To keep the total momentum of the system zero, the gun must move backward with an equal and opposite momentum. This backward motion of the gun is called recoil.

Since the gun has a much larger mass than the bullet, its recoil velocity is much smaller.

Hence, a gun recoils to conserve momentum when the bullet moves forward.

Consider a particle moving in a circular path of radius r. If it makes an angle θ at the center in time t, then:

Angular velocity: ω = θ / t

In the same time, the particle travels an arc length s on the circular path.

Arc length is related to angle by: s = rθ

Linear velocity is distance travelled per unit time: v = s / t

Substitute s = rθ into v = s/t:

v = (rθ) / t

Since ω = θ / t, we get:

v = rω

This is the relation between linear velocity and angular velocity.

Thus, linear speed = radius × angular speed.

1 rpm means one revolution per minute. One revolution = 2π radians, and one minute = 60 seconds.

So

1 rpm = (2π rad) / (60 s) = (π / 30) rad/s

Numerically,

1 rpm ≈ 0.10472 rad/s

Centripetal force is the force required to keep a body moving in a circular path. It always acts towards the centre of the circle and is responsible for changing the direction of the object’s velocity.

Formula:

F = mv² / r

where:
m = mass of the body
v = linear speed
r = radius of the circular path

It can also be written as: F = m r ω²

Given:
Mass, m = 5 kg
Initial velocity, u = 20 m/s
Final velocity, v = 40 m/s
Time, t = 10 s

Acceleration is:

a = (v - u) / t = (40 - 20) / 10 = 2 m/s²

Using Newton’s second law, F = ma:

F = 5 × 2 = 10 N

Therefore, the applied force is 10 N.

Given:
Mass of gun, M = 10 kg
Mass of bullet, m = 50 g = 0.05 kg
Velocity of bullet, v = 400 m/s

Initially, both gun and bullet are at rest, so total momentum = 0.

Using conservation of momentum:

M · V + m · v = 0

Recoil velocity of gun (V):

V = - (m · v) / M

Substitute values:

V = - (0.05 × 400) / 10

V = - 2 / 10 = -0.2 m/s

Recoil velocity of the gun = 0.2 m/s (backward).

Given:
Radius of wheel, r = 0.3 m
Time for one rotation, T = 10 s

One full rotation = 2π radians.

Angular speed:

ω = (2π) / T = (2π) / 10

ω = 0.628 rad/s (approximately)

Therefore, the angular speed of the bicycle wheel is approximately 0.628 rad/s.

Given:
Radius of Ferris wheel, r = 25 m
Time for one rotation, T = 2 minutes = 120 s

Angular speed:

ω = (2π) / T = (2π) / 120 = π/60 rad/s

Tangential speed is:

v = rω = 25 × (π/60)

v ≈ 25 × 0.05236 = 1.309 m/s

Therefore, the tangential speed of the passenger is approximately 1.31 m/s.

Given:
Radius of circular track, r = 50 m
Speed of car, v = 20 m/s

Centripetal acceleration is given by:

ac = v² / r

Substitute values:

ac = (20)² / 50 = 400 / 50 = 8 m/s²

Therefore, the centripetal acceleration of the car is 8 m/s².

Given:
Velocity of car, v = 50 m/s
Radius of curvature, r = 10 m
Mass of person, m = 60 kg

Centrifugal force (same magnitude as centripetal force):

F = m v² / r

Substitute values:

F = 60 × (50)² / 10

F = 60 × 2500 / 10

F = 60 × 250 = 15000 N

Therefore, the centrifugal force experienced is 15,000 N.

Given:
Radius of curve, r = 120 m
Speed of train, 36 km/h = 10 m/s

For banking of rails:

tanθ = v² / (r g)

Substitute values:

tanθ = 10² / (120 × 9.8)

tanθ = 100 / 1176 ≈ 0.085

Angle of banking:

θ = tan⁻¹(0.085) ≈ 4.86°

Therefore, the required angle of banking is approximately 4.86°.

To prevent skidding on a banked road (neglecting friction), the required speed is:

v = √(r g tan θ)

Given:
Radius, r = 100 m
Banking angle, θ = 20°
Acceleration due to gravity, g = 9.8 m/s²

Step 1 — Calculate tan 20°

tan 20° ≈ 0.3639702343

Step 2 — Substitute into the formula

v = √(100 × 9.8 × 0.3639702343)

v = √(356.8626039) ≈ 18.886 m/s

Step 3 — Convert to km/h (optional)

v ≈ 18.886 × 3.6 ≈ 68.0 km/h

Final answer: v ≈ 18.89 m/s ≈ 68.0 km/h