A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of 50° to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find a) the magnitude of the normal reaction of the plane on P, b) the coefficient of friction between P and the plane. c) Determine whether P moves, justifying your answer.

Photo AI

A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 7 - 2014 - Paper 1

Question 7

A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an an... show full transcript

Worked Solution & Example Answer:A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 7 - 2014 - Paper 1

Step 1

a) the magnitude of the normal reaction of the plane on P

96%

114 rated

Answer

To find the normal reaction, we start by resolving the forces acting on the particle. The weight of the particle P can be calculated as:

$W = mg = 2.7 ext{ kg} imes 9.8 ext{ m/s}^2 = 26.46 ext{ N}$

We resolve this weight into two components: one perpendicular to the inclined plane and one parallel to it. The component acting perpendicular to the plane is given by:

$W_{ot} = W imes rac{1}{ ext{cos}(40°)}$ .

The vertical component of the force of 15 N acting at 50° to the plane can also be resolved as:

$F_{ot} = 15 imes ext{sin}(50°)$ .

Using equilibrium, the normal reaction R can be expressed as:

$R = W_{ot} - F_{ot}$ ,

Substituting the values gives us:

$R = 26.46 ext{ N} - (15 imes ext{sin}(50°)) \, ext{N}$ .

Calculating these values leads to the normal reaction of the plane on P.

Step 2

b) the coefficient of friction between P and the plane

99%

104 rated

Answer

To find the coefficient of friction (μ), we use the equation relating frictional force (F_f) to the normal reaction:

$F_f = μR$ .

The frictional force that balances the component of weight parallel to the plane can be expressed as:

$F_f = W_{ ext{parallel}} - F_{15N_{ ext{horizontal}}}$ ,

with

$W_{ ext{parallel}} = W imes ext{sin}(40°)$ .

We can solve for μ:

$μ = \frac{F_f}{R}$ ,

Substituting the calculated values of R and the corresponding F_f will help us find the coefficient of friction.

Step 3

c) Determine whether P moves, justifying your answer

96%

101 rated

Answer

To determine whether P will move, we assess the net force along the inclined plane. If the total downhill force exceeds the total frictional force, then P will move.

The total downhill force can be calculated using:

$F_{down} = W imes ext{sin}(40°) + 15 imes ext{cos}(50°)$ ,

and the friction force as described in part b. If

$F_{down} > F_f$ ,

then particle P will slide down the plane. If they are equal or if friction is greater, P remains stationary. A numerical answer can be calculated based on previously found values.

A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 7 - 2014 - Paper 1

Worked Solution & Example Answer:A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal - Edexcel - A-Level Maths Mechanics - Question 7 - 2014 - Paper 1

a) the magnitude of the normal reaction of the plane on P

b) the coefficient of friction between P and the plane

c) Determine whether P moves, justifying your answer

Join the A-Level students using SimpleStudy...