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Chapter 6: Problem 24

A brick of mass \(1.0 \mathrm{kg}\) slides down an icy roof inclined at \(30.0^{\circ}\) with respect to the horizontal. (a) If the brick starts from rest, how fast is it moving when it reaches the edge of the roof $2.00 \mathrm{m}$ away? Ignore friction. (b) Redo part (a) if the coefficient of kinetic friction is $0.10.

Short Answer

Expert verified
Question: A brick is initially at rest on an inclined roof with an angle of inclination \(\theta\), mass m, and distance s to the edge of the roof. Calculate the final velocity of the brick in (a) a scenario without friction, and (b) a scenario with friction and a coefficient of kinetic friction \(\mu\). Answer: (a) Without friction: \(v_{1} = \sqrt{2gs\sin{\theta}}\) (b) With friction: \(v_{2} = \sqrt{2gs(\sin{\theta} - \mu\cos{\theta})}\)

Step by step solution

01

Identify forces

To start solving the problem, list down the forces acting on the brick. Along the inclined surface, we have two forces acting: 1. Gravitational force (mg) acting vertically downward 2. Frictional force (if applicable) opposite to the direction of movement Step 2: Resolve the forces
02

Resolve forces

Since the forces acting on the brick are in different directions, it's important to resolve them into components parallel to and perpendicular to the inclined surface. The parallel component of gravitational force is given by \[f_{g_\parallel} = mg\sin{\theta}\] and the perpendicular component can be calculated as \[f_{g_\perpendicular} = mg\cos{\theta}\]. Step 3: Calculate net force (a) without friction and (b) with friction
03

Calculate net force

(a) In the scenario without friction, the net force acting on the brick is equal to the parallel component of gravitational force: \[F_{net} = f_{g_\parallel} = mg\sin{\theta}\]. (b) In the scenario with friction, the net force is equal to the difference between parallel component of gravitational force and frictional force: \[F_{net} = f_{g_\parallel} - f_{friction}\], where the frictional force can be computed as \[f_{friction} = \mu f_{g_\perpendicular}\] and \(\mu\) is the coefficient of kinetic friction. Step 4: Use the second law of motion to find the acceleration
04

Find acceleration

According to the second law of motion, the net force acting on an object is equal to the product of its mass and acceleration: \[F_{net} = ma\]. Calculate the acceleration (\(a\)) for both scenarios by dividing the net force by the mass of the brick. Step 5: Use the equations of motion to find the final velocity
05

Calculate final velocity

The equations of motion can be used to find the final velocity. Since the initial velocity (\(u\)) is given as 0, we can use the formula \[v^2 - u^2 = 2as\], where \(v\) - final velocity \(u\) - initial velocity (0) \(a\) - acceleration (calculated in step 4) \(s\) - distance to the edge of the roof Solve for the final velocity (\(v\)) for both scenarios.

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