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. 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.
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:
- Perpendicular to the ramp: Here, ( m = 20 , ext{kg} ), ( g = 9.8 , ext{ms}^{-2} ), and ( heta = 25° ). Therefore,
- Parallel to the ramp:
-
Applying Newton's Second Law: The net force equation along the ramp can be written as: Where:
- ( T = 230 , ext{N} )
- ( F_{ ext{friction}} = , ext{μ} R = , ext{μ} mg , ext{cos}(25°) )
Thus, substituting into the equation:
-
Solve for : After substitution and simplification, we can find ( ext{μ} ) Rearranging to solve for ( ext{μ} ): 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.
Answer
Using the kinematic equation for uniformly accelerated motion, where:
- ( u = 0 ) (initial velocity)
- ( a = 1.2 , ext{ms}^{-2} )
- ( t = 3.8 , ext{s} )
Substituting these values into the equation: = \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).
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.