A block with mass m1 = 9.1 kg is on an incline with an angle θ = 33° with respect to the horizontal. For the first question there is no friction between the incline and the block.
1.) There is a massless rope that pulls horizontally to prevent the block from moving. What is the minimum tension in the rope?
2.) Now a new block is attached to the string above the first block on the ramp. The new block is made of a different material and has a coefficient of static friction μ = 0.97. What is the minimum mass of the new block to keep the system from accelerating?
Friction- Mass on an Incline?
2018-02-07 2:11 pm
回答 (3)
2018-02-07 2:46 pm
Take g = 9.81 m/s²
(The answers may be slightly different from below when g = 9.80 m/s² is taken.)
1.)
Minimum tension of the rope
= Component along the inclined plane of the gravitational force acting on the block
= m₁ g sinθ
= 9.1 × 9.81 × sin33° N
= 48.6 N
2)
When the new block (say, mass = m₂) starts accelerating :
Friction = Components along the inclined plane of the gravitational force acting on the two blocks
(m₂ g cosθ) × μ = m₁ g sinθ + m₂ g sinθ
m₂ μ cosθ = m₁ sinθ + m₂ sinθ
m₂ μ cosθ - m₂ sinθ = m₁ sinθ
m₂ (μ cosθ - sinθ) = m₁ sinθ
m₂ = m₁ sinθ / (μ cosθ - sinθ)
Minimum mass of new block, m₂ = 9.1 × sin33° / (0.97 × cos33° - sin33°) kg = 18.4 kg
Alternative method :
Friction = Component along the inclined plane of the gravitational force acting on the new block + Minimum tension of the rope
(m₂ g cosθ) × μ = m₂ g sinθ + 48.6
m₂ g μ cosθ = m₂ g sinθ + 48.6
m₂ g μ cosθ - m₂ g sinθ = 48.6
m₂ g (μ cosθ - sinθ) = 48.6
m₂ = 48.6 / [g (μ cosθ - sinθ)]
Minimum mass of new block, m₂ = 48.6 / [9.81 × (0.97 × cos33° - sin33°)] kg = 18.4 kg
(The answers may be slightly different from below when g = 9.80 m/s² is taken.)
1.)
Minimum tension of the rope
= Component along the inclined plane of the gravitational force acting on the block
= m₁ g sinθ
= 9.1 × 9.81 × sin33° N
= 48.6 N
2)
When the new block (say, mass = m₂) starts accelerating :
Friction = Components along the inclined plane of the gravitational force acting on the two blocks
(m₂ g cosθ) × μ = m₁ g sinθ + m₂ g sinθ
m₂ μ cosθ = m₁ sinθ + m₂ sinθ
m₂ μ cosθ - m₂ sinθ = m₁ sinθ
m₂ (μ cosθ - sinθ) = m₁ sinθ
m₂ = m₁ sinθ / (μ cosθ - sinθ)
Minimum mass of new block, m₂ = 9.1 × sin33° / (0.97 × cos33° - sin33°) kg = 18.4 kg
Alternative method :
Friction = Component along the inclined plane of the gravitational force acting on the new block + Minimum tension of the rope
(m₂ g cosθ) × μ = m₂ g sinθ + 48.6
m₂ g μ cosθ = m₂ g sinθ + 48.6
m₂ g μ cosθ - m₂ g sinθ = 48.6
m₂ g (μ cosθ - sinθ) = 48.6
m₂ = 48.6 / [g (μ cosθ - sinθ)]
Minimum mass of new block, m₂ = 48.6 / [9.81 × (0.97 × cos33° - sin33°)] kg = 18.4 kg
2018-02-07 7:30 pm
1)
must be
T cos@ > m g sin@
T > m g tan@
T > 9.1*9.8*tan33 = 57.9 N
2)
(m1 + m2) a = (m1 + m2) g sin@ - u (m2 g cos@ + T sin@)
to avoid the system from accelerating [a = 0]
must be
friction force >= driving force
u (m2 g cos@ + T sin@) >= (m1 + m2) g sin@
then
u m2 g cos@ - m2 g sin@ > = m1 g sin@ - T sin@
m2 > = [sin@ (m1 g - T)] / [g (u cos@ - sin@)]
put the numbers and get the slution [6.46 kg]
must be
T cos@ > m g sin@
T > m g tan@
T > 9.1*9.8*tan33 = 57.9 N
2)
(m1 + m2) a = (m1 + m2) g sin@ - u (m2 g cos@ + T sin@)
to avoid the system from accelerating [a = 0]
must be
friction force >= driving force
u (m2 g cos@ + T sin@) >= (m1 + m2) g sin@
then
u m2 g cos@ - m2 g sin@ > = m1 g sin@ - T sin@
m2 > = [sin@ (m1 g - T)] / [g (u cos@ - sin@)]
put the numbers and get the slution [6.46 kg]
2018-02-07 2:28 pm
1)
Self-motive force Fm = m*g*sin 33 = 9.1*9.806*0.545 = 48.63 N
Opponent force F = Fm/cos 33 = m*g*sin 33/cos 33 = 9.1*9.806*tan 33 = 57.95 N
2)
Self-motive force Fm = m*g*sin 33 = 9.1*9.806*0.545 = 48.63 N
Fo = Fm = mx*g*cos 33*μs
mx = 48.63/(9.806*0.839*0.97) = 6.09 kg
Self-motive force Fm = m*g*sin 33 = 9.1*9.806*0.545 = 48.63 N
Opponent force F = Fm/cos 33 = m*g*sin 33/cos 33 = 9.1*9.806*tan 33 = 57.95 N
2)
Self-motive force Fm = m*g*sin 33 = 9.1*9.806*0.545 = 48.63 N
Fo = Fm = mx*g*cos 33*μs
mx = 48.63/(9.806*0.839*0.97) = 6.09 kg
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