Answer to Question #217449 in Classical Mechanics for Help

Question

1) A 40.0 kg box, initially at rest, is pulled through 12.0 m along a horizontal surface. A 75.0 N pulling force is applied at 25.0° from the horizontal. If the frictional force is 14.2 N:

a. How much work was done by the applied force on the box?

b. How much work was done by the frictional force?

c. What is the net work?

d. What is the final speed of the box?

2) A machine has an output power of 13.0 kW. How long would it take for the machine to raise a 4000 kg load through a height of 2.4 m?

3) A 4.0 kg ball is swung at a constant speed in a vertical circle of radius 2.5 m. If the ball completes one revolution in 3.0 s, what is the tension in the rope at the top of the circle?

Answer 1

1)

(a)

$$W_{appl}=F_{appl}dcos\theta=75\ N\cdot12\ m\cdot cos25^{\circ}=816\ J.$$

(b)

$$W_{fr}=F_{fr}dcos\theta=14.2\ N\cdot12\ m\cdot cos180^{\circ}=-170.4\ J.$$

(c)

$$W_{net}=W_{appl}+W_{fr}=816\ J+(-170.4\ J)=645.6\ J.$$

(d)

$$KE_f=W_{net},$$ $$\dfrac{1}{2}mv_f^2=W_{net},$$ $$v_f=\sqrt{\dfrac{2W_{net}}{m}}=\sqrt{\dfrac{2\cdot645.6\ J}{40\ kg}}=5.68\ \dfrac{m}{s}.$$

2)

$$P=\dfrac{W}{t}=\dfrac{mgh}{t},$$ $$t=\dfrac{mgh}{P}=\dfrac{4000\ kg\cdot9.8\ \dfrac{m}{s^2}\cdot2.4\ m}{13\cdot10^3\ W}=7.2\ s.$$

3)

$$T=F_c-F_g,$$ $$T=\dfrac{mv^2}{r}-mg,$$ $$T=\dfrac{m(\dfrac{2\pi r}{T})^2}{r}-mg,$$ $$T=\dfrac{4\pi^2rm}{T^2}-mg,$$ $$T=\dfrac{4\pi^2\cdot2.5\ m\cdot4.0\ kg}{(0.3\ s)^2}-4.0\ kg\cdot9.8\ \dfrac{m}{s^2}=4347\ N.$$