Classical Mechanics - Acceleration

A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x) = βx⁻ⁿ where β and n are constants and x is the position of the particle. What is the acceleration of the particle as a function of x?

A. −nβ²x⁻²ⁿ⁻¹
B. −nβ²x⁻ⁿ⁻¹
C. −nβ²x⁻ⁿ
D. −β²x⁻ⁿ⁺¹
E. −β²x⁻²ⁿ⁺¹

(GR9677 #44)

Solution:

a = dv/dt = (dv/dx) (dx/dt) =  (dv/dx) v

Given: v(x) = βx⁻ⁿ
dv/dx =  −nβx⁻ⁿ⁻¹

Thus, a = (−nβx⁻ⁿ⁻¹) βx⁻ⁿ = −nβ²x⁻²ⁿ⁻¹

Answer: A

Sound and Wave - Wave Phenomena

Consider a particle moving without friction on a rippled surface, as shown above. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos (kx). If the particle starts at x = 0 with a speed v in the x direction, for what values of v will the particle stay on the surface at all times?

A.
B.
C.
D.
E. 

(GR9677 #83)

Solution:

Gravity pulls the particle down. 
Particle will stay on the surface if  a_max = g

For a wave, a_max = ω²A = ω²d
(A = d = amplitude)

Velocity of the particle: v =  λ / T 

From the picture: λ = 2π/k

v = 2π/kT  = ω/k
v² = ω²/k²
ω² = v²k²

a_max = v²k²d = g 
v² = g/k²d
v = √(g/k²d)

Answer: D 

Classical Mechanics - Circular Motion

A rigid cylinder rolls at constant speed without slipping on top of a horizontal plane surface. The acceleration of a point on the circumference of the cylinder at the moment when the point touches the plane is

A. directed forward
B. directed backward
C. directed up
D. directed down
E. zero

(GR9277 #40)

Solution:

Constant speed → no tangential acceleration → Uniform Circular Motion



When the point touches the plane,

 is directed up

Answer: C

Lab Methods - Standard Deviation

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are σ_m and σ_a respectively, then σ_F/F is

A.

B.

C.

D.

E.

(GR9277 #48)

Solution:

Multiplying two quantities with uncertainty:

u = x ⋅ y



For F = ma:



Answer: C

Classical Mechanics - 1D Vertical Motion

A rock is thrown vertically upward with initial speed v₀. Assume a friction force proportional to –v, where v is the velocity of the rock, and neglect the buoyant force exerted by air. Which of the following is correct?

1. The acceleration of the rock is always equal to g.
2. The acceleration of the rock is equal to g only at the top of the flight.
3. The acceleration of the rock is always less than g.
4. The speed of the rock upon return to its starting point is v₀.
5. The rock can attain a terminal speed greater than v₀ before it returns to its starting point.

(GR8677 #01)

Solution:

There is a friction force:

• acceleration is not constant → (A) and (C) FALSE
• energy is not conserved so it's initial and final speed is not the same → (D) FALSE
• frictional force slows down the object, so its speed at time t has to be less than its initial speed  → (E) FALSE

Answer: B


Math analysis:

Friction force: F_f  = − kv
The equation of motion with friction force: ma = − mg − kv.

a = − g − (kv/m)
(A) FALSE

At the top of the flight, v = 0
a = − g − (k∙0/m) = − g
(B) TRUE

Moving up: v positive
a = − g − (kv/m) = − g − c
a − g
Moving Down: v negative
a = − g − [k(−v)/m] = − g + c
a − g
(C) FALSE

Classical Mechanics - Kinematics

A particle is initially at rest at the top of a curved frictionless track. The x- and y-coordinate of the track are related in dimensionless units by y = x²/4, where the positive y-axis is in the vertical downward direction. As the particle slides down the track, what its tangential acceleration?

A. 0
B. g
C. gx/2
D. gx/√(x²+4)
E. (gx²)/√(x²+16)

(GR8677 #06)

Solution:

A. FALSE.
The particle slides down the curved track → the tangential acceleration is not zero.

B. FALSE.
The particle slides down the curved track, not in the vertical downward direction (the direction of gravity acceleration) → the tangential acceleration is not equal to g.

C. FALSE.
The unit of gx/2 is not the unit of acceleration.

D. TRUE.
The unit of gx/√(x²+4) = g → the unit of acceleration.

E. FALSE.
The unit of (gx²)/√(x²+16) is not the unit of acceleration.

Answer: D

Calculation:

Classical Mechanics - Pendulum

Which of the following best illustrates the acceleration of a pendulum bob at points a through e?

(GR0177 #01)

Solution:

The acceleration of pendulum: a = a_centripetal + a_tangential
a_cent = v²/r = ω²r
a_tan = αr

At equilibrium (position B): ω = constant
α = dω/dt = 0
a = a_cent

At maximum amplitude (position A and C): v = 0
a = a_tan

At other positions:
a = a_cent + a_tan

Answer: C