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:
a) Ffr = -0.18 N
b) a= -1.64 m/s2
c) t = 9.2 s
d) x = 68.7 m.
e) W= -12.4 J
f) Pavg = -1.35 W
g) Pinst = -0.72 W
Explanation:
a)
- While the puck slides across ice, the only force acting in the horizontal direction, is the force of kinetic friction.
- This force is the horizontal component of the contact force, and opposes to the relative movement between the puck and the ice surface, causing it to slow down until it finally comes to a complete stop.
- So, this force can be written as follows, indicating with the (-) that opposes to the movement of the object.
where μk is the kinetic friction coefficient, and Fn is the normal force.
- Since the puck is not accelerated in the vertical direction, and there are only two forces acting on it vertically (the normal force Fn, upward, and the weight Fg, downward), we conclude that both must be equal and opposite each other:
- We can replace (2) in (1), and substituting μk by its value, to find the value of the kinetic friction force, as follows:
b)
- According Newton's 2nd Law, the net force acting on the object is equal to its mass times the acceleration.
- In this case, this net force is the friction force which we have already found in a).
- Since mass is an scalar, the acceleration must have the same direction as the force, i.e., points to the left.
- We can write the expression for a as follows:
c)
- Applying the definition of acceleration, choosing t₀ =0, and that the puck comes to rest, so vf=0, we can write the following equation:
- Replacing by the values of v₀ = 15 m/s, and a = -1.64 m/s2, we can solve for t, as follows:
d)
- From (1), (2), and (3) we can conclude that the friction force is constant, which it means that the acceleration is constant too.
- So, we can use the following kinematic equation in order to find the displacement before coming to rest:
- Since the puck comes to a stop, vf =0.
- Replacing in (7) the values of v₀ = 15 m/s, and a= -1.64 m/s2, we can solve for the displacement Δx, as follows:
e)
- The total work done by the friction force on the object , can be obtained in several ways.
- One of them is just applying the work-energy theorem, that says that the net work done on the object is equal to the change in the kinetic energy of the same object.
- Since the final kinetic energy is zero (the object stops), the total work done by friction (which is the only force that does work, because the weight and the normal force are perpendicular to the displacement) can be written as follows:
f)
- By definition, the average power is the rate of change of the energy delivered to an object (in J) with respect to time.
- If we choose t₀=0, replacing (9) as ΔE, and (6) as Δt, and we can write the following equation:
g)
- The instantaneous power can be deducted from (10) as W= F*Δx, so we can write P= F*(Δx/Δt) = F*v (dot product)
- Since F is constant, the instantaneous power when v=4.0 m/s, can be written as follows: