Q: A projectile is launched with initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the maximum height reached by the projectile?

$$h_{max} = \frac{v_0$^2 \sin^2$ \theta}{2g}$$

Question 1

A solid sphere and a solid cylinder, both with the same mass $$m$$ and radius $$R$$, are released from rest at the top of an inclined plane of height $$h. $$Assuming no slipping, which object reaches the bottom first, and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia relative to its mass and radius.

Question 2

A block of mass $$m$$ slides down a frictionless incline of height $$h. $$What is its speed at the bottom of the incline?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 3

Which of the following is the correct expression for the work done by a variable force $$F(x)$$ over a displacement from $$x_1 to$$ $$x_2$$?

Accepted Answer

$$W = \int_{x_1$}^{x_2$} F(x) \, dx$$

Question 4

A mass $$m$$ attached to a spring oscillates with angular frequency $$\omega. $$What is the period $$T of $$oscillation?

Accepted Answer

$$T = \frac{2\pi}{\omega}$$

Question 5

A car of mass $$m is $$moving at speed $$v. $$What is its kinetic energy?

Accepted Answer

$$K = \frac{1}{2}mv^2$

Question 6

A force $$F$$ acts at an angle $$	heta to $$the direction of displacement $$d. $$What is the work done by the force?

Accepted Answer

$$W = Fd \cos 	heta$$

Question 7

Which of the following is the correct formula for the moment of inertia of a solid sphere of mass $$M$$ and radius $$R$$ about its center?

Accepted Answer

$$I = \frac{2}{5}MR^2$

Question 8

A projectile is launched with initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the maximum height reached by the projectile?

Accepted Answer

$$h_{max} = \frac{v_0$^2 \sin^2$ 	heta}{2g}$$

Question 9

A ball is dropped from a height $$h$$ above the ground. Neglecting air resistance, what is the time it takes to reach the ground?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}}$$

Question 10

A block of mass $$m is $$attached to a spring with spring constant $$k$$ and oscillates on a frictionless surface. What is the angular frequency of oscillation?

Accepted Answer

$$\omega = \sqrt{\frac{k}{m}}$$

Question 11

A uniform rod of length $$L$$ and mass $$M$$ rotates about an axis through one end perpendicular to its length. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 12

A force $$F is $$applied to a mass $$m$$ for a time $$\Delta t. $$What is the change in momentum of the mass?

Accepted Answer

$$\Delta p = F \Delta t$$

Question 13

A satellite orbits the Earth in a circular orbit of radius $$r. $$What is the speed required for a stable orbit?

Accepted Answer

$$v = \sqrt{\frac{GM}{r}}$$

Question 14

A block slides across a rough horizontal surface and comes to rest due to friction. Which principle best explains the loss of mechanical energy?

Accepted Answer

The work done by friction converts mechanical energy into thermal energy.

Question 15

A ball is thrown vertically upward with initial speed $$v_0. $$What is the total time the ball spends in the air before returning to the thrower's hand?

Accepted Answer

$$t = \frac{2v_0}{g}$$

Question 16

A car rounds a flat curve of radius $$r at $$constant speed $$v. $$What is the minimum coefficient of static friction $$\mu$$ required to prevent slipping?

Accepted Answer

$$\mu = \frac{v^2$}{rg}$$

Question 17

A pendulum of length $$L$$ swings with small amplitude. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}}$$

Question 18

A block of mass $$m is $$attached to a horizontal spring and oscillates with amplitude $$A. $$What is the maximum speed of the block?

Accepted Answer

$$v_{max} = A \omega$$

Question 19

A uniform disk of mass $$M$$ and radius $$R$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2}MR^2$

Question 20

A ball is thrown horizontally from a cliff of height $$h$$ with initial speed $$v_0. $$How far from the base of the cliff does the ball land?

Accepted Answer

$$x = v_0$ \sqrt{\frac{2h}{g}}$$

PHYS 1001H Formula Sheet: Kinematics, Dynamics, Work & Energy, Oscillations, and Rotational Motion

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