Required value of initial speed of the bullet be ( 4M/m)√(gL).
Given parameters:
Mass of the bullet =m.
Mass of the bob of the pendulum = M.
speed of the bullet before collision = v
Speed of the bullet after collision = v/2.
Length of the pendulum stiff rod = L.
Let speed transmitted to the pendulum be u.
Using principle of conservation of momentum:
mv = Mu + mv/2
⇒ Mu = mv/2
⇒ u = (m/M)v/2
We know that: to make the bob over the top of the trajectory without falling backward in its circular path, required speed be = √(4gL). [ where g = acceleration due to gravity]
To be minimum initial speed the bullet must have in order for the pendulum bob to just barely swing through a complete vertical circle:
u = √(4gL)
⇒ (m/M)v/2 = √(4gL)
⇒ v =( 4M/m)√(gL).
Hence, minimum required speed of the bullet be ( 4M/m)√(gL).
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With the use of electric force formula, the direction and magnitude of the net force exerted on the point charge q3 are 9.9 x
N and 66 degrees
ELECTRIC FORCE (F)
F = 
Where K = 9 x
N
/
The distance between
and
can be calculated by using Pythagoras theorem.
d = 
d = 46.7 cm = 0.467 m
For force
, substitute all the parameters into the formula above
= (9 x
x 3 x 1)/
= 2.7 x
/0.218
= 1.24 x
N
For force
, substitute all the parameters into the formula above
= (9 x
x 3 x 4)/
= 1.08 x
/0.1089
= 9.92 x
N
For force
, substitute all the parameters into the formula above
= (9 x
x 3 x 2)/
= 5.4 x
/0.1089
= 4.96 x
N
Summation of forces on Y component will be
=
-
Sin 45
= 9.92 x
- 1.24 x
Sin 45
= 9.04 x
N
Summation of forces on X component will be
=
-
Cos 45
= 4.96 x
- 1.24 x
Sin 45
= 4.08 x
N
Net Force = 
Net force = 
Net force = 9.9 x
N
The direction will be
Tan ∅ =
/
Tan ∅ = 9.04 x
/ 4.08 x 
Tan ∅ = 2.216
∅ =
(2.216)
∅ = 65.7 degrees
Therefore, the direction and magnitude of the net force exerted on the point charge q3 are 9.9 x
N and 66 degrees approximately.
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The three longest wavelengths for the standing waves on a 264-cm long string that is fixed at both ends are:
- 5.2 meters.
- 2.6 meters.
- 1.7meters.
Given data:
Length of the fixed string = 264cms = 2.64 meters
The wavelength for standing waves is given by:
λ = 2L/n
where,
- λ is the wavelength
- L is the length of the string
For n = 1,
= 5.2 meters
For n = 2,
= 2.6 meters
For n = 3,
= 1.7 meters
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Answer:
t = 5.89 s
Explanation:
To calculate the time, we need the radius of the pulley and the radius of the sphere which was not given in the question.
Let us assume that the radius of the pulley (
) = 0.4 m
Let the radius of the sphere (r) = 0.5 m
w = angular speed = 150 rev/min = (150 × 2π / 60) rad/s = 15.708 rad/s
Tension (T) = 20 N
mass (m) = 3 kg each


Substituting values:
