Q: A proton and an electron are placed 1.0 m apart in vacuum. What is the magnitude of the electric force between them? (Use $$k = 8.99 \times 10^9$ \ \mathrm{Nm}^2/\mathrm{C}^2$$, $$e = 1.60 \times 10$$^{-19}$$ \ \mathrm{C})$$

$$F = \dfrac{k e^2$}{r^2$} = \dfrac{8.99 \times 10^9$ \times (1.60 \times 10$$^{-19}$$)^2$$}{$$1^2$$} = 2.30 \times $$10^{-28}$$ \ \mathrm{N}$

Q: A 2 kg object is dropped from a height of 5 m. What is its gravitational potential energy relative to the ground? (Use $$g = 9.80 \ \mathrm{m/s}^2)$$

$$U_G = mgy = 2 \times 9.80 \times 5 = 98 \ \mathrm{J}$$

Q: A spring with spring constant $$k = 100 \ \mathrm{N/m} is $$compressed by $$\Delta s = 0.2 \ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

$$U_S = \dfrac{1}{2} k (\Delta s)^2$ = \dfrac{1}{2} \times 100 \times (0.2)^2$ = 2 \ \mathrm{J}$$

Q: A uniform rod of mass $$m = 3 \ \mathrm{kg}$$ and length $$L = 2 \ \mathrm{m}$$ rotates about one end. What is its moment of inertia?

$$I = \dfrac{1}{3} m L^2$ = \dfrac{1}{3} \times 3 \times 2^2$ = 4 \ \mathrm{kg \cdot m^2$}$$

Q: A block slides across a rough surface with kinetic friction coefficient $$\mu_k = 0.3$$ and normal force $$n = 50 \ \mathrm{N}. $$What is the work done by friction after the block moves $$d = 4 \ \mathrm{m}$$?

$$W = -f_k d = -\mu_k n d = -0.3 \times 50 \times 4 = -60 \ \mathrm{J}$$

Q: A particle of mass $$m = 0.5 \ \mathrm{kg} is $$moving at $$v = 3 \ \mathrm{m/s}. $$What is its kinetic energy?

$$K = \dfrac{1}{2} m v^2$ = \dfrac{1}{2} \times 0.5 \times 3^2$ = 2.25 \ \mathrm{J}$$

Q: A charge $$q = 2 \times 10$$^{-6}$$ \ \mathrm{C} is $$placed in an electric field $$E = 500 \ \mathrm{N/C}. $$What is the force on the charge?

$$F_E = q E = 2 \times 10$$^{-6}$$ \times 500 = 1 \times 10$$^{-3}$$ \ \mathrm{N}$$

Q: A cylinder has a linear charge density $$\lambda = 1 \times 10$$^{-6}$$ \ \mathrm{C/m}. $$What is the electric field at a distance $$r = 0.1 \ \mathrm{m}$$ from the axis? (Use $$\$$epsilon_0$$ = 8.85 \times 10$$^{-12}$$ \ \mathrm{F/m})$$

$$E = \dfrac{\lambda}{2 \pi \$$epsilon_0$$ r} = \dfrac{1 \times 10$$^{-6}$$}{2 \pi \times 8.85 \times 10$$^{-12}$$ \times 0.1} \approx 1.8 \times 10^4$ \ \mathrm{N/C}$$

Question 1

A proton and an electron are placed 1.0 m apart in vacuum. What is the magnitude of the electric force between them? (Use $$k = 8.99 	imes 10^9$ \ \mathrm{Nm}^2/\mathrm{C}^2$$, $$e = 1.60 	imes 10$$^{-19}$$ \ \mathrm{C})$$

Accepted Answer

$$F = \dfrac{k e^2$}{r^2$} = \dfrac{8.99 	imes 10^9$ 	imes (1.60 	imes 10$$^{-19}$$)^2$$}{$$1^2$$} = 2.30 	imes $$10^{-28}$$ \ \mathrm{N}$

Question 2

A 2 kg object is dropped from a height of 5 m. What is its gravitational potential energy relative to the ground? (Use $$g = 9.80 \ \mathrm{m/s}^2)$$

Accepted Answer

$$U_G = mgy = 2 	imes 9.80 	imes 5 = 98 \ \mathrm{J}$$

Question 3

A spring with spring constant $$k = 100 \ \mathrm{N/m} is $$compressed by $$\Delta s = 0.2 \ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

Accepted Answer

$$U_S = \dfrac{1}{2} k (\Delta s)^2$ = \dfrac{1}{2} 	imes 100 	imes (0.2)^2$ = 2 \ \mathrm{J}$$

Question 4

A particle moves in a circle of radius $$r = 0.5 \ \mathrm{m}$$ with a speed $$v = 2 \ \mathrm{m/s}. $$What is its centripetal acceleration?

Accepted Answer

$$a_r = \dfrac{v^2$}{r} = \dfrac{2^2$}{0.5} = 8 \ \mathrm{m/s}^2$$

Question 5

A uniform rod of mass $$m = 3 \ \mathrm{kg}$$ and length $$L = 2 \ \mathrm{m}$$ rotates about one end. What is its moment of inertia?

Accepted Answer

$$I = \dfrac{1}{3} m L^2$ = \dfrac{1}{3} 	imes 3 	imes 2^2$ = 4 \ \mathrm{kg \cdot m^2$}$$

Question 6

A block slides across a rough surface with kinetic friction coefficient $$\mu_k = 0.3$$ and normal force $$n = 50 \ \mathrm{N}. $$What is the work done by friction after the block moves $$d = 4 \ \mathrm{m}$$?

Accepted Answer

$$W = -f_k d = -\mu_k n d = -0.3 	imes 50 	imes 4 = -60 \ \mathrm{J}$$

Question 7

A particle of mass $$m = 0.5 \ \mathrm{kg} is $$moving at $$v = 3 \ \mathrm{m/s}. $$What is its kinetic energy?

Accepted Answer

$$K = \dfrac{1}{2} m v^2$ = \dfrac{1}{2} 	imes 0.5 	imes 3^2$ = 2.25 \ \mathrm{J}$$

Question 8

A charge $$q = 2 	imes 10$$^{-6}$$ \ \mathrm{C} is $$placed in an electric field $$E = 500 \ \mathrm{N/C}. $$What is the force on the charge?

Accepted Answer

$$F_E = q E = 2 	imes 10$$^{-6}$$ 	imes 500 = 1 	imes 10$$^{-3}$$ \ \mathrm{N}$$

Question 9

A cylinder has a linear charge density $$\lambda = 1 	imes 10$$^{-6}$$ \ \mathrm{C/m}. $$What is the electric field at a distance $$r = 0.1 \ \mathrm{m}$$ from the axis? (Use $$\$$epsilon_0$$ = 8.85 	imes 10$$^{-12}$$ \ \mathrm{F/m})$$

Accepted Answer

$$E = \dfrac{\lambda}{2 \pi \$$epsilon_0$$ r} = \dfrac{1 	imes 10$$^{-6}$$}{2 \pi 	imes 8.85 	imes 10$$^{-12}$$ 	imes 0.1} \approx 1.8 	imes 10^4$ \ \mathrm{N/C}$$

Question 10

A disk of radius $$R$$ has a uniform surface charge density $$\eta. $$What is the electric field just above the center of the disk?

Accepted Answer

$$E = \dfrac{\eta}{2 \$$epsilon_0$$}$$

Question 11

A particle moves under a force $$F = 10 \ \mathrm{N}$$ for a displacement $$\Delta r = 3 \ \mathrm{m} at an $$angle $$	heta = 60^\circ to $$the direction of force. What is the work done?

Accepted Answer

$$W = F \Delta r \cos 	heta = 10 	imes 3 	imes \cos 60^\circ = 10 	imes 3 	imes 0.5 = 15 \ \mathrm{J}$$

Question 12

A mass $$m = 1 \ \mathrm{kg}$$ attached to a spring with $$k = 16 \ \mathrm{N/m}$$ oscillates with amplitude $$A = 0.5 \ \mathrm{m}. $$What is the total energy of the system?

Accepted Answer

$$E = \dfrac{1}{2} k A^2$ = \dfrac{1}{2} 	imes 16 	imes (0.5)^2$ = 2 \ \mathrm{J}$$

Question 13

A ball is thrown vertically upward with initial velocity $$v_0 = 10$$ \ \mathrm{m/s}$$. How high does it rise? (Use g = 9.80$$ \ \mathrm{m/s}^2$$)

Accepted Answer

$$h = \dfrac{v_0$^2}{2g} = \dfrac{10^2$}{2 	imes 9.80} = 5.10 \ \mathrm{m}$$

Question 14

A solid sphere of mass $$m = 2 \ \mathrm{kg}$$ and radius $$R = 0.5 \ \mathrm{m}$$ rolls without slipping. What is its moment of inertia about its center?

Accepted Answer

$$I = \dfrac{2}{5} m R^2$ = \dfrac{2}{5} 	imes 2 	imes (0.5)^2$ = 0.2 \ \mathrm{kg \cdot m^2$}$$

Question 15

A particle of mass $$m = 1 \ \mathrm{kg}$$ moves with velocity $$v = 4 \ \mathrm{m/s} in $$the $$\hat{i}$$ direction. What is its momentum vector?

Accepted Answer

$$\vec{p} = m \vec{v} = 1 	imes 4 \hat{i} = 4 \hat{i} \ \mathrm{kg \cdot m/s}$$

Question 16

A torque $$	au = 5 \ \mathrm{N \cdot m} is $$applied to a disk with moment of inertia $$I = 2 \ \mathrm{kg \cdot m^2$}. $$What is its angular acceleration?

Accepted Answer

$$\alpha = \dfrac{	au}{I} = \dfrac{5}{2} = 2.5 \ \mathrm{$$rad/s^2$$}$$

Question 17

A vector $$\vec{A} = 2 \hat{i} + 3 \hat{j}$$ and $$\vec{B} = 4 \hat{i} + 1 \hat{j}. $$What is $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$\vec{A} \cdot \vec{B} = 2 	imes 4 + 3 	imes 1 = 8 + 3 = 11$$

Question 18

A particle moves along the $$x-$$axis with position $$x(t) = 3t^2$ + 2t + 1. $$What is its velocity at $$t = 2 \ \mathrm{s}$$?

Accepted Answer

$$v = \dfrac{dx}{dt} = 6t + 2 \Rightarrow v(2) = 6 	imes 2 + 2 = 14 \ \mathrm{m/s}$$

Question 19

A simple harmonic oscillator has $$k = 9 \ \mathrm{N/m}$$ and $$m = 1 \ \mathrm{kg}. $$What is its angular frequency?

Accepted Answer

$$\omega = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{9}{1}} = 3 \ \mathrm{rad/s}$$

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