Q: A 5 kg box is moving to the right at 3 m/s. What is its kinetic energy?

$$K = \frac{1}{2}mv^2$ = \frac{1}{2}(5)(3^2$) = 22.5\ \text{J}$$

Q: A force $$F(x) = 3x^2$ is applied to a box as it moves from x = 2$$\ \text{m}$$ to x = 5$$\ \text{m}$$. What is the total work done by this force?

$$W = \$$int_2$$^5 3x^2$ dx = [x^3$]_2^5$ = 125 - 8 = 117\ \text{J}$$

Q: A 10 kg suitcase is pulled with a 50 N force at an angle of $$37^\circ$$ above the horizontal for 20 m. What is the work done by the horizontal component of the force?

$$W = F_x d = (50 \cos 37^\circ)(20) = (39.9)(20) = 798\ \text{J}$$

Q: A 75 kg hiker climbs a 1000 m mountain. What is the work done by gravity during the hike?

$$W_g = -mg\Delta y = -(75)(9.8)(1000) = -735,000\ \text{J}$$

Q: A 2 kg rock is thrown vertically upward with an initial velocity of 15 m/s. What is the initial kinetic energy of the rock?

$$K = \frac{1}{2}mv^2$ = \frac{1}{2}(2)(15^2$) = 225\ \text{J}$$

Q: A box weighing 50 N is pulled horizontally with 60 N for 8 m. The coefficient of kinetic friction is $$\mu_k = 0.7. $$What is the work done by friction?

$$W_{f_k} = -f_k d = -\mu_k N d = -(0.7)(50)(8) = -280\ \text{J}$$

Question 1

Which of the following statements best describes the Work-Energy Theorem?

Accepted Answer

The net work done on an object is equal to its change in kinetic energy: $$W_{	ext{net}} = \Delta K = K_f - K_i$$

Question 2

A 5 kg box is moving to the right at 3 m/s. What is its kinetic energy?

Accepted Answer

$$K = \frac{1}{2}mv^2$ = \frac{1}{2}(5)(3^2$) = 22.5\ 	ext{J}$$

Question 3

A force $$F(x) = 3x^2$ is applied to a box as it moves from x = 2$$\ 	ext{m}$$ to x = 5$$\ 	ext{m}$$. What is the total work done by this force?

Accepted Answer

$$W = \$$int_2$$^5 3x^2$ dx = [x^3$]_2^5$ = 125 - 8 = 117\ 	ext{J}$$

Question 4

A 10 kg suitcase is pulled with a 50 N force at an angle of $$37^\circ$$ above the horizontal for 20 m. What is the work done by the horizontal component of the force?

Accepted Answer

$$W = F_x d = (50 \cos 37^\circ)(20) = (39.9)(20) = 798\ 	ext{J}$$

Question 5

A 75 kg hiker climbs a 1000 m mountain. What is the work done by gravity during the hike?

Accepted Answer

$$W_g = -mg\Delta y = -(75)(9.8)(1000) = -735,000\ 	ext{J}$$

Question 6

A 2 kg rock is thrown vertically upward with an initial velocity of 15 m/s. What is the initial kinetic energy of the rock?

Accepted Answer

$$K = \frac{1}{2}mv^2$ = \frac{1}{2}(2)(15^2$) = 225\ 	ext{J}$$

Question 7

A box weighing 50 N is pulled horizontally with 60 N for 8 m. The coefficient of kinetic friction is $$\mu_k = 0.7. $$What is the work done by friction?

Accepted Answer

$$W_{f_k} = -f_k d = -\mu_k N d = -(0.7)(50)(8) = -280\ 	ext{J}$$

Question 8

Calculate the dot product of $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = \hat{i} + 2\hat{j}.$$

Accepted Answer

$$\vec{A} \cdot \vec{B} = (2)(1) + (3)(2) = 2 + 6 = 8$$

Question 9

A spring with force constant $$k = 500\ 	ext{N/m} is $$compressed by 2 m. What is the work done BY the spring?

Accepted Answer

$$W_s = -\frac{1}{2}kx^2$ = -\frac{1}{2}(500)(2^2$) = -1000\ 	ext{J}$$

Question 10

A 1000 kg elevator is lifted at constant speed to a height of 100 m in 50 seconds. What is the average power supplied by the motor?

Accepted Answer

$$P_{avg} = \frac{W}{\Delta t} = \frac{mgh}{t} = \frac{(1000)(9.8)(100)}{50} = 19,600\ 	ext{W}$$

Question 11

A force $$F(x) = 2e$$^{-2x}$$ is applied to a can as it moves from x = 0.5$ m to x = 1.5$ m. What is the work done by this force?

Accepted Answer

$$W = \int_{0.5}^{1.5} 2e$$^{-2x}$$ dx = [-e$$^{-2x}$$]_{0.5}^{1.5} = -e$$^{-3}$$ + e$$^{-1}$$ \approx 0.368 - 0.050 = 0.318\ 	ext{J}$$

Question 12

A 15 kg block is accelerated at $$2.0\ 	ext{m/s}^2$$ along a frictionless surface, with its speed increasing from $$10\ 	ext{m/s} to$$ $$30\ 	ext{m/s}. $$What is the change in the block’s kinetic energy?

Accepted Answer

$$\Delta K = \frac{1}{2}m(v_f^2$ - v_i^2$) = \frac{1}{2}(15)(30^2$ - 10^2$) = \frac{1}{2}(15)(900 - 100) = 7.5 	imes 800 = 6000\ 	ext{J}$$

Question 13

Which of the following is true about the dot product of two vectors?

Accepted Answer

The dot product is a scalar and equals $$|\vec{A}| |\vec{B}| \cos 	heta$$

Question 14

A 4 kg box has an initial speed of 6 m/s at point A and a speed of 10 m/s at point B. How much work was done on the box from A to B?

Accepted Answer

$$W = \Delta K = \frac{1}{2}m(v_f^2$ - v_i^2$) = \frac{1}{2}(4)(100 - 36) = 2 	imes 64 = 128\ 	ext{J}$$

Question 15

A 1.0 m-long spring takes 200 J to compress to 70 cm. How much additional work is required to compress it from 70 cm to 50 cm?

Accepted Answer

$$W = \frac{1}{2}k(x_2$^2 - x_1$^2)$$, where $$x_1 = 0.3$ m, x_2$ = 0.5 m$$. $$W = \frac{1}{2}k(0.5^2$ - 0.3^2$). $$Find k from $$200 = \frac{1}{2}k(0.3^2$)$$, $$k = 200/(0.09/2) = 4444 N/m$$. $$W = \frac{1}{2}(4444)(0.25 - 0.09) = 333.3 J$$

Question 16

A 7 kg crate slides 2.5 m down a ramp inclined at $$26^\circ. $$The coefficient of friction is 0.36. What is the work done by friction?

Accepted Answer

$$W_{f_k} = -\mu_k mg \cos 	heta \cdot d = -0.36 	imes 7 	imes 9.8 	imes \cos 26^\circ 	imes 2.5 \approx -55.6\ 	ext{J}$$

Work, Kinetic Energy, and Applications (Ch 09 Study Notes)

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