In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal. The bottom of the ramp is at the point O. A crate of mass 20 kg is at rest at the point A on the ramp. The crate is pulled up the ramp using a rope attached to the crate. Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp. 19 (a) The tension in the rope is 230 N The crate accelerates up the ramp at 1.2 m/s² Find the coefficient of friction between the crate and the ramp. 19 (b) (i) The crate takes 3.8 seconds to reach the top of the ramp. Find the distance OA. 19 (b) (ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).

Photo AI

In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal - AQA - A-Level Maths Pure - Question 19 - 2022 - Paper 2

Question 19

$In-this-question-use-$g-=-9.8-\,--ext{ms}^{-2}$--A-rough-wooden-ramp-is-10-metres-long-and-is-inclined-at-an-angle-of-25°-above-the-horizontal-AQA-A-Level Maths Pure-Question 19-2022-Paper 2.png$

In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal. The bottom of the ram... show full transcript

Worked Solution & Example Answer:In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal - AQA - A-Level Maths Pure - Question 19 - 2022 - Paper 2

Step 1

19 (a) The tension in the rope is 230 N. The crate accelerates up the ramp at 1.2 m/s². Find the coefficient of friction between the crate and the ramp.

96%

114 rated

Answer

To find the coefficient of friction, we first need to resolve the forces acting on the crate along the ramp and perpendicular to the ramp.

Resolve Weight: The weight of the crate can be resolved into two components:
- Parallel to the ramp:
  $W_{ ext{parallel}} = mg \, ext{sin} \, heta$
- Perpendicular to the ramp: $W_{ ext{perpendicular}} = mg \, ext{cos} \, heta$ Here, ( m = 20 , ext{kg} ), ( g = 9.8 , ext{ms}^{-2} ), and ( heta = 25° ). Therefore, $W_{ ext{parallel}} = 20 imes 9.8 imes ext{sin}(25°)$ $W_{ ext{perpendicular}} = 20 imes 9.8 imes ext{cos}(25°)$
Applying Newton's Second Law: The net force equation along the ramp can be written as: $T - W_{ ext{parallel}} - F_{ ext{friction}} = ma$ Where:
- ( T = 230 , ext{N} )
- ( F_{ ext{friction}} = , ext{μ} R = , ext{μ} mg , ext{cos}(25°) )
Thus, substituting into the equation: $230 - (20 imes 9.8 imes ext{sin}(25°)) - ext{μ}(20 imes 9.8 \times ext{cos}(25°)) = 20(1.2)$
Solve for $ext{μ}$ : After substitution and simplification, we can find ( ext{μ} ) Rearranging to solve for ( ext{μ} ): $230 - 196 ext{sin}(25°) - 20 ext{μ} (9.8 ext{cos}(25°)) = 24$ This leads to an expression to calculate ( ext{μ} ) yielding ( ext{μ} , ext{≈} , 0.69 ).

Step 2

19 (b) (i) The crate takes 3.8 seconds to reach the top of the ramp. Find the distance OA.

99%

104 rated

Answer

Using the kinematic equation for uniformly accelerated motion, $s = ut + \frac{1}{2} at^2$ where:

( u = 0 ) (initial velocity)
( a = 1.2 , ext{ms}^{-2} )
( t = 3.8 , ext{s} )

Substituting these values into the equation: $s = 0(3.8) + \frac{1}{2}(1.2)(3.8^2)$ = \frac{1}{2}(1.2)(14.44) \approx 8.664 , ext{m} $$ Therefore, the distance OA is approximately ( 10 , ext{m} - 8.664 , ext{m} = 1.336 , ext{m} ).

Step 3

19 (b) (ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).

96%

101 rated

Answer

One assumption made is that the crate is treated as a particle, meaning we neglect its dimensions and consider all its mass concentrated at a point. This simplifies the dynamics involved in the motion up the ramp.

In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal - AQA - A-Level Maths Pure - Question 19 - 2022 - Paper 2

Worked Solution & Example Answer:In this question use $g = 9.8 \, ext{ms}^{-2}$ A rough wooden ramp is 10 metres long and is inclined at an angle of 25° above the horizontal - AQA - A-Level Maths Pure - Question 19 - 2022 - Paper 2

19 (a) The tension in the rope is 230 N. The crate accelerates up the ramp at 1.2 m/s². Find the coefficient of friction between the crate and the ramp.

19 (b) (i) The crate takes 3.8 seconds to reach the top of the ramp. Find the distance OA.

19 (b) (ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).

Join the A-Level students using SimpleStudy...