5.00-kg particle starts from the origin at time zero. Its velocity as a function of time is given by v =6t^2 i + 2t j where v is
2 answers:
The concept of derivatives and integrals allows to find the results for the questions are the motion of a particle where the speed depends on time are:
a)the position is: r = 2 t³ i + t² j
b) the position of the body on the y-axis is a parabola and on the x-axis it is a cubic function
c) The acceleration is: a = 12 t i + 2 j
d) the force is: F = 60 t i + 10 j
e) the torque is: τ = 40 t³ k^
f) tha angular momentum is: L = 4t³ - 6 t² k^
g) The kinetic energy is: K = 2 m t² (9t² +1)
h) The power is: P = 2m (36 t³ + 2t)
Kinematics studies the movement of bodies, looking for relationships between position, speed and acceleration.
a) They indicate the function of speed.
v = 6 t² i + 2t j
Ask the function of the position. The velocity is defined by the variation of the position with respect to time
v =
dr = v dt
we substitute and integrate.
∫ dr = ∫ (6 t² i + 2t j) dt
r - 0 =
r = 2 t³ i + t² j
b) The position of the body on the y axis is a parabola and on the x axis it is a cubic function.
c) Acceleration is defined as the change in velocity with time.
a =
a = 12 t i + 2 j
d) Newton's second law states that force is proportional to mass times the body's acceleration.
F = ma
F = m (12 t i + 2 j)
F = 5 12 t i + 2 j
F = 60 t i + 10 j
e) Torque is the vector product of the force and the distance to the origin.
τ = F x r
The easiest way to write these expressions is to solve for the determinant.
τ = (60t t² - 2t³ 10) k
τ = 40 t³ k ^
f) Angular momentum
L = r x p
L =rx (mv)
L = m (rxv)
The easiest way to write these expressions is to solve for the determinant.
L = (4t³ - 6 t²) k
g) The kinetic energy is
K = ½ m v²
K = ½ m (6 t² i + 2t j) ²
K = m 18 t⁴ + 2t²
K = 2 m t² (9t² +1)
h) Power is work per unit time
P = dW / dt
The relationship between work and kinetic energy
W = ΔK
P =
p = 2m (36 t³ + 2t)
In conclusion with the concept of derivatives and integrals we can find the results for the questions are the motion of a particle where the speed depends on time are:
a) The position is: r = 2 t³ i + t² j
b) The position of the body on the y-axis is a parabola and on the x-axis it is a cubic function
c) The acceleration is: a = 12 t i + 2 j
d) The force is: F = 60 t i + 10 j
e) The torque is: τ = 40 t³ k^
f) The angular momentum is: L = 4t³ - 6 t² k^
g) The kinetic energy is: K = 2 m t² (9t² +1)
h) The power is: P = 2m (36 t³ + 2t)
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Answer:
The question is incomplete. However, I believe, it is asking for the acceleration of the elevator. This is 3.16 m/s².
Explanation:
By Hooke's law,
F is the force on a spring, k is the spring constant and e is the extension or compression.
From the question,
This is the force on the mass suspended on the spring. Its acceleration, a, is given by
This acceleration is more than the acceleration due to gravity, g = 9.8 m/s². Hence the elevator must be moving up with an acceleration of
12.96 - 9.8 m/s² = 3.16 m/s²