Q: A person of mass $m$ sits on a swing attached to two cables, one horizontal and one at $40.0^\circ$ above the horizontal. The swing moves in a horizontal circle of radius $r$ at a constant speed $v$. What is the tension $T_1$ in the horizontal cable?

$T_1 = m v^2 / r - m g \tan 40^\circ$

Q: A block of mass $m_1$ slides down a frictionless ramp and a block of mass $m_2$ moves upward, after both blocks have moved a distance $s$. How does the total work done on $m_1$ compare to the total work done on $m_2$?

Less than the total work done on $m_2$, but not zero.

Q: A block of mass $m$ is pushed up a ramp with a constant force $F$ parallel to the ramp surface. The coefficient of kinetic friction between the box and the ramp surface is $\mu_k$. What is the magnitude of the acceleration of the box?

$a = \dfrac{F - m g \sin \theta - \mu_k m g \cos \theta}{m}$

Q: A spring has a force law $F_x = kx + bx^2 + cx^3$, where $k = 100$ N/m, $b = 70$ N/m$^2$, and $c = -1200$ N/m$^3$. How much work must be done to stretch this spring by $0.050$ m from its unstretched length?

$W = \dfrac{1}{2} k x^2 + \dfrac{1}{3} b x^3 + \dfrac{1}{4} c x^4$

Q: A 20.0 N block moves 75.0 cm to the right and a 12.0 N block moves 75.0 cm downward, connected by a string passing over a frictionless pulley. If there is no friction between the table and the 20.0 N block, what is the total work done on each block?

The total work done on each block is equal in magnitude but opposite in sign.

Q: A block of mass $m$ slides across a rough section of floor. The coefficient of friction is not constant: it starts at $\mu_0$ and increases linearly with distance past point $P$, reaching a value of $\mu_f$ at distance $d$ past $P$. What is the work done by friction as the block slides from $P$ to $P + d$?

$W_f = -\dfrac{1}{2} (\mu_0 + \mu_f) m g d$

Q: A person of mass $m$ is sitting in a swing attached to two cables, one horizontal and one at $40.0^\circ$ above the horizontal. The swing moves in a horizontal circle of radius $r$ at a constant speed $v$. What is the tension $T_2$ in the angled cable?

$T_2 = \dfrac{m g}{\cos 40^\circ} + \dfrac{m v^2}{r \sin 40^\circ}$

Q: A block of mass $m$ is pulled up a rough ramp of angle $\theta$ with a constant force $F$ parallel to the ramp. The coefficient of kinetic friction is $\mu_k$. What is the minimum force $F$ required to move the block up the ramp at constant speed?

$F = m g \sin \theta + \mu_k m g \cos \theta$

Q: A block of mass $m$ is released from rest and slides down a frictionless incline of angle $\theta$. After sliding a distance $d$, it reaches the bottom and encounters a rough patch with coefficient of kinetic friction $\mu_k$. How far will the block slide on the rough patch before coming to rest?

$s = \dfrac{2 d \sin \theta}{\mu_k}$

Q: A block of mass $m$ is attached to a spring with force law $F_x = kx + bx^2 + cx^3$. Is it easier to stretch or compress this spring? Explain in terms of the dependence of $F_x$ on $x$.

It is easier to compress the spring because $F_x$ increases more slowly for negative $x$.

Question 1

An elevator is being lifted at a constant speed by a steel cable attached to an electric motor. There is no air resistance, nor is there any friction between the elevator and the walls of the elevator shaft. What is the upward force exerted on the elevator by the cable?

Accepted Answer

Equal to the force of gravity.

Question 2

A block of mass $m$ is placed at rest on a frictionless incline of angle $\alpha$. It is released and slides a distance $s$ along the incline before reaching the floor, where it encounters a kinetic friction force $f$ and stops after sliding a distance $d$. What is the net work done on the block during its entire motion?

Accepted Answer

Zero.

Question 3

A cable attached to a car holds the car at rest on a frictionless ramp (angle $\alpha < 90^\circ$). The ramp exerts a normal force on the car. How does the magnitude of the normal force compare to the weight of the car?

Accepted Answer

$n < w$

Question 4

A person of mass $m$ sits on a swing attached to two cables, one horizontal and one at $40.0^\circ$ above the horizontal. The swing moves in a horizontal circle of radius $r$ at a constant speed $v$. What is the tension $T_1$ in the horizontal cable?

Accepted Answer

$T_1 = m v^2 / r - m g 	an 40^\circ$

Question 5

A block of mass $m_1$ slides down a frictionless ramp and a block of mass $m_2$ moves upward, after both blocks have moved a distance $s$. How does the total work done on $m_1$ compare to the total work done on $m_2$?

Accepted Answer

Less than the total work done on $m_2$, but not zero.

Question 6

A block of mass $m$ is pushed up a ramp with a constant force $F$ parallel to the ramp surface. The coefficient of kinetic friction between the box and the ramp surface is $\mu_k$. What is the magnitude of the acceleration of the box?

Accepted Answer

$a = \dfrac{F - m g \sin 	heta - \mu_k m g \cos 	heta}{m}$

Question 7

A spring has a force law $F_x = kx + bx^2 + cx^3$, where $k = 100$ N/m, $b = 70$ N/m$^2$, and $c = -1200$ N/m$^3$. How much work must be done to stretch this spring by $0.050$ m from its unstretched length?

Accepted Answer

$W = \dfrac{1}{2} k x^2 + \dfrac{1}{3} b x^3 + \dfrac{1}{4} c x^4$

Question 8

A 20.0 N block moves 75.0 cm to the right and a 12.0 N block moves 75.0 cm downward, connected by a string passing over a frictionless pulley. If there is no friction between the table and the 20.0 N block, what is the total work done on each block?

Accepted Answer

The total work done on each block is equal in magnitude but opposite in sign.

Question 9

A block of mass $m$ slides across a rough section of floor. The coefficient of friction is not constant: it starts at $\mu_0$ and increases linearly with distance past point $P$, reaching a value of $\mu_f$ at distance $d$ past $P$. What is the work done by friction as the block slides from $P$ to $P + d$?

Accepted Answer

$W_f = -\dfrac{1}{2} (\mu_0 + \mu_f) m g d$

Question 10

A person of mass $m$ is sitting in a swing attached to two cables, one horizontal and one at $40.0^\circ$ above the horizontal. The swing moves in a horizontal circle of radius $r$ at a constant speed $v$. What is the tension $T_2$ in the angled cable?

Accepted Answer

$T_2 = \dfrac{m g}{\cos 40^\circ} + \dfrac{m v^2}{r \sin 40^\circ}$

Question 11

A block of mass $m$ is released from rest and slides down a frictionless incline of angle $	heta$. What is its speed after sliding a distance $d$ down the incline?

Accepted Answer

$v = \sqrt{2 g d \sin 	heta}$

Question 12

A block of mass $m$ is pulled up a rough ramp of angle $	heta$ with a constant force $F$ parallel to the ramp. The coefficient of kinetic friction is $\mu_k$. What is the minimum force $F$ required to move the block up the ramp at constant speed?

Accepted Answer

$F = m g \sin 	heta + \mu_k m g \cos 	heta$

Question 13

A block of mass $m$ is attached to a rope and moves upward with a velocity magnitude that varies in time according to $v(t) = (2.00 	ext{ m/s}) + (0.500 	ext{ m/s}^2) t^2$. What is the tension in the rope when the velocity of the box is $9.00$ m/s?

Accepted Answer

$T = m g + m a$

Question 14

A block of mass $m$ slides across a rough section of floor. The coefficient of friction is $\mu_k$. If the block starts with speed $v_0$ and comes to rest after sliding a distance $d$, what is the value of $\mu_k$?

Accepted Answer

$\mu_k = \dfrac{v_0^2}{2 g d}$

Question 15

A block of mass $m$ is released from rest and slides down a frictionless incline of angle $	heta$. After sliding a distance $d$, it reaches the bottom and encounters a rough patch with coefficient of kinetic friction $\mu_k$. How far will the block slide on the rough patch before coming to rest?

Accepted Answer

$s = \dfrac{2 d \sin 	heta}{\mu_k}$

Question 16

A block of mass $m$ is pulled by a force $F$ at an angle $\alpha$ above the horizontal across a rough surface with coefficient of kinetic friction $\mu_k$. What is the normal force $N$ exerted by the surface on the block?

Accepted Answer

$N = m g - F \sin \alpha$

Question 17

A block of mass $m$ is attached to a spring with force law $F_x = kx + bx^2 + cx^3$. Is it easier to stretch or compress this spring? Explain in terms of the dependence of $F_x$ on $x$.

Accepted Answer

It is easier to compress the spring because $F_x$ increases more slowly for negative $x$.

Physics 21: Forces, Work, and Energy (Chapters 4–6) – Study Guide

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