Finding the work done by a block

In summary, the conversation discussed how to calculate the work done by friction on a block sliding down an inclined plane with a rough surface. The force of gravity, normal force, and friction were considered, and the net force, acceleration, velocity, and distance traveled were calculated using kinematic equations. The final expression for the work done on the block by friction at any given time t is μ * M * g * cos(θ) * (h - (1/2) * g * t^2), where h is the initial height of the block, t is the time, μ is the coefficient of friction, and physical constants are used where appropriate. The distance traveled equation is based on the general kinematic (SUVAT)
  • #1
HeavenWind
2
0
Homework Statement
The problem is: A block with mass M is placed onto a rough inclined plane. After the block is released, it immediately begins to accelerate down the ramp due to gravity. The coefficient of friction for this ramp is μ. A stopwatch measures time t in seconds after the block is released at height h.

Derive an expression that represents the work done on the block by friction at time t in terms of t, h, θ, m, μ, and physical constants as appropriate.
Relevant Equations
F_normal = M * g * cos(θ)
F_gravity = M * g
F_friction = μ * F_normal
F_net / M
d = h - (1/2) * g * t^2.
v = √(2 * a * (h - (1/2) * g * t^2))
We want to figure out how much work friction does on a block as it slides down an inclined plane with a rough surface.

we find the force due to gravity that pulls the block down the ramp, that's found by M * g * sin(θ),

The normal force on the block is given by M * g * cos(θ).

The force of friction acting on the block is μ * F_normal

The net force acting on the block is found by subtracting the force of friction from the force due to gravity.

The acceleration of the block down the ramp is given by the net force divided by the mass of the block.

We find the velocity of the block at any given time by multiplying the acceleration by the time.

We find the distance traveled by the block at any given time by using kinematic equations.

Finally, we find the work done by friction by multiplying the force of friction by the distance traveled by the block. It's given by μ * M * g * cos(θ) * (h - (1/2) * g * t^2), where h is the initial height of the block, t is the time, and physical constants are used where appropriate.
 
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  • #2
To be more specific, here is what my friend did, not sure if it is correct:

To derive the expression for the work done on the block by friction at time t, we need to consider the forces acting on the block.

Let's assume that the inclined plane makes an angle θ with the horizontal. The force due to gravity acting on the block is given by:

F_gravity = M * g * sin(θ)

where g is the acceleration due to gravity.

The normal force acting on the block is given by:

F_normal = M * g * cos(θ)

The force of friction acting on the block is given by:

F_friction = μ * F_normal

where μ is the coefficient of friction.

The net force acting on the block is the sum of the force due to gravity and the force of friction:

F_net = F_gravity - F_friction

= M * g * sin(θ) - μ * M * g * cos(θ)

= M * g * (sin(θ) - μ * cos(θ))

The acceleration of the block down the ramp is given by:

a = F_net / M

= g * (sin(θ) - μ * cos(θ))

The velocity of the block at time t is given by:

v = a * t

= g * (sin(θ) - μ * cos(θ)) * t

The distance traveled by the block at time t is given by:

d = h - (1/2) * g * t^2

The work done on the block by friction at time t is given by:

W_friction = F_friction * d

= μ * F_normal * (h - (1/2) * g * t^2)

= μ * M * g * cos(θ) * (h - (1/2) * g * t^2)

Therefore, the expression that represents the work done on the block by friction at time t in terms of t, h, θ, m, μ, and physical constants as appropriate is:

W_friction = μ * M * g * cos(θ) * (h - (1/2) * g * t^2)
 
  • #3
The distance traveled is not vertical and is not due to motion with acceleration g.
 
  • #4
HeavenWind said:
The distance traveled by the block at time t is given by:

d = h - (1/2) * g * t^2
Which general kinematic (SUVAT) equation is that based on? How are the terms defined in it?
 
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  • #5
The distance traveled by the block at time t is given by:

d = h - (1/2) * g * t^2

haruspex said:
Which general kinematic (SUVAT) equation is that based on? How are the terms defined in it?

Yes - you were all good down to this point. Just use the acceleration down the slope which you had correctly worked out.
 

1. What is the definition of work done by a block?

The work done by a block is the product of the force applied to the block and the displacement of the block in the direction of the force.

2. How is the work done by a block calculated?

The work done by a block is calculated by multiplying the force applied to the block by the displacement of the block in the direction of the force. This can be represented by the equation W = F * d.

3. What is the unit of measurement for work done by a block?

The unit of measurement for work done by a block is joules (J). This is equivalent to 1 newton-meter (N*m).

4. Can the work done by a block be negative?

Yes, the work done by a block can be negative. This occurs when the force applied to the block is in the opposite direction of the displacement of the block. This indicates that work is being done against the motion of the block.

5. How does friction affect the work done by a block?

Friction can decrease the work done by a block by opposing the motion of the block. This means that the force applied to the block must overcome the force of friction in order to move the block, resulting in less work being done.

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