Physics Problem - Potential Energy

The potential energy is given by the function U(x) = kx⁻¹, where k is a constant, and variable x is the position, the coordinate. (a) Derive the function of the coordinate x, which describes the force produced by this potential energy.

F(x) = - d U(x) / dx

F(x) = - d kx⁻¹ / dx = kx⁻²

F(x) = kx⁻²

(b) Let this potential energy is obtained by the body with the mass of 10kg. The constant is known: k = -3.98 × 10¹⁵ Jm.
Find the force on this body when it is located atx = 6.38 × 10⁶ m?

F(x) = kx⁻² = -3.98 × 10¹⁵ Jm / ( 6.38 × 10⁶ m)²

The code for the Google calculator: (-3.98E15J*m) / ( 6.38E6m )^2

Google calculator's result: -97.7781272 newtons

Problem's answer: F = -97.8 N or 97.8 N towards x = 0

How Much Work Is Done by the Spring on the Mass?

A horizontal spring having constant k is attached to a block having mass M. The mass is oscillating freely on a frictionless table with amplitude A. As the mass travels from the position of the maximum stretch to half the position of maximum stretch, how much work is done by the spring on the mass?

Given Data

k = 50.0 N/m

M= 0.50 kg

A= 15.0 cm

xₒ=A

x=A/2

W=?

Solution

U=½kx²

Uₒ=½kxₒ²

Uₒ-U=W

W=½kxₒ²-½kx²=½k(xₒ²-x²)

W=½k(A²-(½A)²)=½k(A²-(½A)²)=½k(A²-¼A²)=½k(¾A²)=⅜kA²

W=⅜·(50 N/m)·(0.15m)²

Code for the Google Calculator: 3/8*(50 N/m)*(0.15m)^2

Google Calculator's result: 0.421875 joules

Answer:   W=0.42J

Quiz 5, Problem - Find Average Power

What is the average power output of an engine when a car of mass M accelerates uniformly (a=const) from rest to a final speed V over a distance d on flat, level ground? Ignore energy lost due to friction and air resistance. Derive a formula for average power Pₐ in terms of variables: the mass M, the final speed V, and the distance d.
Pₐ=Pₐ(M,V,d) ?

Pₐ=W/t

K-Kₒ=W

Kₒ=0

K=½MV²

Pₐ=½MV²/t

V=at

d=½at²= ½Vt

t=2d/V

Pₐ=½MV²/(2d/V) = ¼MV³/d

A train with the mass M accelerates uniformly from the rest to the speed V when passing through a distance d

Problem from Quiz 5

A train with the mass M accelerates uniformly from the rest to the speed V when passing through a distance d.
What is its kinetic energy
when it passing through a distance d₁ (d₁ < d)?

Given Data: M = 15,000 kg

Vₒ = 0

V = 15 m / s

d = 150 m

d₁ = 75 m

K₁ = ?

Solution:

K(kinetic energy) = Kₒ(kinetic energy initial) + W(work) = W

W = Fx

K = W = Fd = ½MV²

F = ½MV² / d

K₁ = W₁ = Fd₁ = ½MV² d₁ / d

K₁ = ½(15,000 kg)(15 m/s)² (75m) / (150m) = 843750 J

Google Calculator Code:

0.5*(15000 kg)*(15 m/s)^2*(75m) / (150m)

Answer: 840 kJ

Physics Problem - Average Power

7.36. The electric motor of a 2-kg train accelerates the train from rest to 1 m/s in 1 s. Find the average power delivered to the train during the acceleration.

Solution

Given Data	Useful formulas
M=2kg V₀=0 V=1m/s t=1s pₐ=?	K=½MV² K₀=0 pₐ=W/t W= K-K₀	pₐ=W/t=(K-K₀)/t=½MV²/t pₐ = ½MV²/t pₐ= ½ (2kg) (1m/s)² /1s = =1 kg m/s²∙m/s=1N∙m/s=1J/s=1W pₐ=1W

7.40. A 1000-kg elevator starts from rest.
It moves upward for 4 s with constant acceleration until it reaches its cruising speed of 2 m/s.

(a) What is the average power of the elevator motor during this period?

(b) What is the motor power when the elevator moves at its cruising speed?

Given Data	Useful formulas
M=1000kg (a) V₀=0 V=2m/s t=4s pₐ=? (b) V=const=2m/s	K=½MV² K₀=0 U=Mgh pₐ=W/t W= =(K+U)-(K₀+ U₀) h=V₀t+at²/2 a=(V-V₀)/t p = F V =  = MgV	W= (K+U)-(K₀+ U₀)= K+U= =½MV²+Mgh=½MV²+Mgh h=at²/2=(V/t)∙t²/2=Vt/2 W= ½MV²+MgVt/2 pₐ=W/t=½MV²/t+MgV/2 or pₐ=W/t=M(V/t+g)∙½V ½MV²/t=½(1000kg)(2m/s)²/4s=500W MgV/2=1000kg∙10m/s²∙2m/s /2=10000W pₐ=10500W=10.5kW or pₐ=1000kg∙(2 m/s / 4 s +10 m/s²)∙½∙2 m/s= =10.5kW p=MgV=1000kg∙10m/s²∙2m/s=20000W=20kW

Deadline

The announcement for CityTech students:

The deadline for online quizzes 2,3,4,5, and 6 is October 6 (Thursday), 4:00 PM.

After the deadline, these quizzes will be closed for records, and correct answers will be displayed.

A car moving in the x direction has acceleration aₓ, that varies with time as shown in the figure

A car moving in the x direction has acceleration aₓ, that varies with time as shown in the figure. pic.twitter.com/xLnJUtQIyp

— Physics I (@PHY210) September 28, 2016