AP Physics C Mechanics - 2.4 Newton’s First Law- Exam Style questions- FRQs

Newton’s First Law AP  Physics C Mechanics FRQ

Unit 2: Force and Translational Dynamics

Weightage : 20-15%

AP Physics C Mechanics Exam Style Questions – All Topics

Question

Blocks of mass $m$ and $2m$ are connected by a light string and placed on a frictionless inclined plane that makes an angle $\theta$ with the horizontal, as shown in Figure 1 above. Another light string connecting the block of mass $m$ to a hanging sphere of mass $M$ passes over a pulley of negligible mass and negligible friction. The entire system is initially at rest and in equilibrium.
 
(a) On the dots below that represent the block of mass $m$ and the sphere of mass $M$, draw and label the forces (not components) that act on each of the objects shown. Each force must be represented by a distinct arrow starting on and pointing away from the dot.
(b) Derive expressions for the magnitude of each of the following. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figures in part (a).
i. The force $T_2$ exerted on the block of mass $m$ by the string. Express your answers in terms of $m$, $\theta$, and physical constants, as appropriate.
ii. The mass $M$ for which the system can remain in equilibrium. Express your answers in terms of $m$, $\theta$, and physical constants, as appropriate.
(c) Now suppose that mass $M$ is large enough to descend and that the sphere reaches the floor before the blocks reach the pulley. Answer the following for the moment immediately after the sphere reaches the floor.
i. Does the tension $T_1$ increase, decrease to a nonzero value, decrease to zero, or stay the same?
_____ Increase _____ Decrease to a nonzero value _____ Decrease to zero _____ Stay the same
ii. Is the velocity of the block of mass $m$ up the ramp, down the ramp, or zero?
_____ Up the ramp _____ Down the ramp _____ Zero
iii. Is the acceleration of the block of mass $m$ up the ramp, down the ramp, or zero?
_____ Up the ramp _____ Down the ramp _____ Zero
(d) Consider the initial setup in Figure 1. Now suppose the surface of the incline is rough and the coefficient of static friction between the blocks and the inclined plane is $\mu_s$. Derive an expression for the minimum possible value of $M$ that will keep the blocks from moving down the incline. Express your answer in terms of $m$, $\mu_s$, $\theta$, and fundamental constants, as appropriate.
(e) The string connecting block $m$ and the sphere of mass $M$ then breaks, and the blocks begin to move from rest down the incline. The lower block starts a distance $d$ from the bottom of the incline, as shown in Figure 1. The coefficient of kinetic friction between the blocks and the inclined plane is $\mu_k$. Derive an expression for the speed of the blocks when the lower block reaches the bottom of the incline. Express your answer in terms of $m$, $d$, $\mu_k$, $\theta$, and fundamental constants, as appropriate.

Most-appropriate topic codes (AP Physics C: Mechanics):

• Topic 2.2 — Forces and Free-Body Diagrams (Part a)
• Topic 2.4 — Newton’s First Law (Parts b, c)
• Topic 2.5 — Newton’s Second Law (Parts b, c, e)
• Topic 2.7 — Kinetic and Static Friction (Parts d, e)
• Topic 3.4 — Conservation of Energy (Part e)
▶️ Answer/Explanation

(a)
For the block of mass $m$:
• Normal force $F_N$ pointing perpendicularly away from the incline.
• Gravitational force $mg$ pointing straight down.
• Tension $T_1$ pointing directly down the incline.
• Tension $T_2$ pointing directly up the incline.
For the sphere of mass $M$:
• Gravitational force $Mg$ pointing straight down.
• Tension $T_2$ pointing straight up.

(b)(i)
Treat the two blocks as a single system of mass $3m$ on the incline.
Apply Newton’s second law for equilibrium ($F_{\text{net}} = 0$):
$T_2 – (m + 2m)g \sin\theta = 0$
$T_2 = 3mg \sin\theta$

(b)(ii)
The sphere of mass $M$ is also in equilibrium.
Apply Newton’s second law ($F_{\text{net}} = 0$):
$T_2 – Mg = 0$
Substitute the expression for $T_2$ from part (i):
$3mg \sin\theta – Mg = 0$
$M = 3m \sin\theta$

(c)(i)
Decrease to zero. Once the sphere $M$ hits the floor, $T_2$ immediately drops to zero. Without an upward pull, the blocks are free to move. Because both blocks have the same acceleration down the frictionless ramp, they don’t pull on each other, causing the tension $T_1$ between them to become slack (zero).

(c)(ii)
Up the ramp. Immediately before the sphere hit the ground, the mass $M$ was descending, meaning the blocks on the incline were moving up the ramp. By inertia, their velocity in that exact instant immediately after remains directed up the ramp.

(c)(iii)
Down the ramp. Without the tension $T_2$ pulling them up, the only force acting along the incline is the component of gravity pulling the blocks downward. This produces a net acceleration directed down the ramp.

(d)
We want the minimum mass $M$ to keep the blocks from sliding down, meaning static friction is acting up the incline to assist $M$.
Apply Newton’s second law for the system along the incline plane ($F_{\text{net}} = 0$):
$Mg + f_{s1} + f_{s2} – (m + 2m)g \sin\theta = 0$
Substitute $f_s = \mu_s F_N$ where $F_N = mg \cos\theta$ for each block:
$Mg + \mu_s(mg \cos\theta) + \mu_s(2mg \cos\theta) – 3mg \sin\theta = 0$
$Mg + 3\mu_s mg \cos\theta = 3mg \sin\theta$
$M = 3m(\sin\theta – \mu_s \cos\theta)$

(e)
First, find the acceleration of the blocks sliding down the incline with kinetic friction.
Apply Newton’s second law ($F_{\text{net}} = m_{\text{total}} a$):
$3mg \sin\theta – f_{k1} – f_{k2} = (3m)a$
Substitute kinetic friction $f_k = \mu_k F_N$ for both blocks:
$3mg \sin\theta – \mu_k(mg \cos\theta) – \mu_k(2mg \cos\theta) = 3ma$
$3mg \sin\theta – 3\mu_k mg \cos\theta = 3ma$
$a = g(\sin\theta – \mu_k \cos\theta)$
Now, use the kinematics equation for an object accelerating from rest ($v_0 = 0$) over a distance $d$:
$v^2 = v_0^2 + 2ad$
$v^2 = 0 + 2g(\sin\theta – \mu_k \cos\theta)d$
$v = \sqrt{2gd(\sin\theta – \mu_k \cos\theta)}$

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