A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s⁻¹ when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp.

A-Level Edexcel Maths Mechanics Forces: A lifeboat slides down a straigh

A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 4 - 2013 - Paper 1

Question 4

A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat i... show full transcript

Worked Solution & Example Answer:A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 4 - 2013 - Paper 1

Step 1

Finding acceleration

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Answer

Using the equation of motion, we start with:

$v^2 = u^2 + 2as$

where:

v = final velocity = 12.6 m/s
u = initial velocity = 0 m/s (released from rest)
s = distance = 50 m
a = acceleration.

Substituting the values:

$12.6^2 = 0^2 + 2a(50)$

Solving for acceleration (a):

$a = \frac{12.6^2}{2 \cdot 50} = 1.5876 \, \text{m/s}^2$

Step 2

Using forces on the ramp

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Answer

The net force acting on the lifeboat along the ramp can be expressed as:

$F = mg \sin(\theta) - f$

Where:

m = mass of the lifeboat = 800 kg
g = acceleration due to gravity = 9.81 m/s²
$\theta$ = angle of inclination = 15°
f = frictional force = $\mu R$ .

The normal force (R) acting on the lifeboat is:

$R = mg \cos(\theta) = 800 \cdot 9.81 \cdot \cos(15°)$

Step 3

Calculating friction and coefficient of friction

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Answer

Substituting in the equations:

$800g \sin(15°) - \mu (800g \cos(15°)) = 800a$

We know $a = 1.5876$ m/s². Plugging values for g:

$800 \cdot 9.81 \cdot \sin(15°) - \mu (800 \cdot 9.81 \cdot \cos(15°)) = 800 \cdot 1.5876$

We can simplify and solve for the coefficient of friction ( $\mu$ ). Rearranging gives us:

$\mu = \frac{800 \cdot 9.81 \sin(15°) - 800 \cdot 1.5876}{800 \cdot 9.81 \cos(15°)}$

A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 4 - 2013 - Paper 1

Worked Solution & Example Answer:A lifeboat slides down a straight ramp inclined at an angle of 15° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 4 - 2013 - Paper 1

Finding acceleration

Using forces on the ramp

Calculating friction and coefficient of friction

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