Download Obtain (i) the velocity and acceleration at any time, (ii)

COLLEGE OF SCIENCES
DEPARTMENT OF MATHEMATICAL AND PHYSICAL SCIENCES
PHY 214 ANALYTICAL MECHANICS
2016/17 SESSION TUTORIAL QUESTIONS
(TO BE SUBMITTED ON/BEFORE THURSDAY 15TH JUNE 2017)
1.
(a)
(b)
(c)
What is rectilinear motion? Hence, for a particle traveling along the 𝑥-axis,
write the expressions which describe its (i) average velocity, (ii)
instantaneous velocity, (iii) average acceleration, and (iv) instantaneous
acceleration.
A particle’s position at any time 𝑡 is given by: 𝑥 = (2𝑡 4 + 5𝑡 2 + 3𝑡 )𝑚.
Obtain (i) the velocity and acceleration at any time, (ii) the position, the
velocity and acceleration at 𝑡 = 2𝑠 𝑎𝑛𝑑 3𝑠, and (iii) the average velocity
and average acceleration between 𝑡 = 2𝑠 𝑎𝑛𝑑 3𝑠.
If the particle in (b) above moves under the influence of a constant force 𝐹
and its initial speed is 𝑉𝑜 , show that the speed of the particle at any position
𝑥 is given by:
2𝐹
𝑉 = √𝑉𝑜2 + ( ) 𝑥
𝑚
where 𝑚 is the mass of the particle.
2.
(a)
(b)
(c)
What is referred to as a freely falling body? Write the differential equation
from Newton’s second law for such body.
An object of mass 𝑚 is thrown vertically upward from the earth’s surface
with speed 𝑉𝑜 . Find (i) the position at any time, (ii) the time taken to reach
the highest point, (iii) the maximum height reached
Given that a uniform force with linear frictional damping acts on mass m in
𝑑𝑣
(b) above, such that its equation of motion is 𝑚 = −𝑚𝑔 − 𝛽𝑉, show that
𝑑𝑡
the position of m at any time is:
𝛽
𝑚
𝑚
𝑚
𝑧 = (𝑉𝑜 + 𝑔) (1 − 𝑒 −𝑚𝑡 ) − 𝑔𝑡
𝛽
𝛽
𝛽
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3.
(a)
(b)
When is a particle said to be in equilibrium?
A particle of mass 𝑚 is suspended in equilibrium by two inelastic strings of
lengths 𝑎 and 𝑏 which are separated by a distance 𝑐 apart as shown below.
Find the tension in each string.
4.
(a)
Show that the magnitude of centripetal acceleration of a particle undergoing
circular motion is:
𝑉2
𝑎=
𝑟
A particle moves so that its position vector is given by 𝑟⃗ = cos 𝜔𝑡 𝑖̂ +
⃗⃗ of the particle
sin 𝜔𝑡 𝑗̂ where 𝜔 is a constant. Show that (𝑎) the velocity 𝑉
is perpendicular to 𝑟⃗, (𝑏) the acceleration 𝑎⃗ is directed toward the origin
⃗⃗ =
and has magnitude proportional to the distance from the origin, (𝑐 ) 𝑟⃗ × 𝑉
𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟.
(b)
5.
6.
(a)
(i)
(ii)
(iii)
(b)
(i)
(ii)
(a)
(b)
Describe a constrained motion?
A particle 𝑃 of mass 𝑚 slides without rolling down a frictionless plane
inclined at angle 𝛼. If it starts from rest at the top of the incline, find (𝑖) the
acceleration, (𝑖𝑖) the velocity, and (𝑖𝑖𝑖) the distance traveled after time 𝑡.
Suppose the incline in (b) has a coefficient of friction 𝜇. Find (i) the
acceleration, (ii) the speed, and (iii) the distance travelled by the particle
after time 𝑡.
(c)
7.
(a)
State and prove the work-energy theorem.
Under what conditions is a force field said to be conservative.
Apply the conditions in (ii) to show that
𝐹⃗ = (𝑦 2 𝑧 3 − 6𝑥𝑧 2 )𝑖̂ + 2𝑥𝑦𝑧 3 𝑗̂ + (3𝑥𝑦 2 𝑧 2 − 6𝑥 2 𝑧)𝑘̂
is a conservative force field
State and prove the principle of conservation of angular momentum
⃗⃗⃗.
Obtain the relationship between Torque 𝜏⃗ and angular momentum Ω
The equation of motion of a simple harmonic oscillator is given by
𝑑2 𝑥
𝑚 2 𝑖̂ = −𝑘𝑥𝑖̂
𝑑𝑡
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(i)
(ii)
(b)
(c)
8.
(a)
(b)
Obtain the general solution to the equation and the boundary
conditions.
Obtain the period (P) and frequency (f) of the motion.
Assuming the harmonic oscillator in (a) above has a damping force
⃗⃗ acting on it. Obtain the solutions for the following cases
𝐹⃗𝐷 = −𝛽𝑉
(𝑖) Over-damped motion (𝑖𝑖) Critically damped motion, and (𝑖𝑖𝑖) Underdamped motion
Show that the system 𝑥̈ + 4𝑥̇ + 3𝑥 = 0 is over-damped.
Distinguish between Constrained motion and Free fall motion
At time 𝑡 = 0 a parachutist having weight of magnitude 𝑚𝑔 is located at
𝑧 = 0 and is travelling vertically downward with speed 𝑉𝑜 . If the force or
air resistance acting on the parachute is proportional to the instantaneous
speed, find
(𝑖) the speed at any time 𝑡 > 0,
(𝑖𝑖) the distance travelled at any time 𝑡 > 0,
(𝑖𝑖𝑖) the acceleration at any time 𝑡 > 0, and
𝑚𝑔
(𝑖𝑣) show that the parachutist approaches a limiting speed given by
⁄𝛽.
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