Finding the work done by a block

[HeavenWind]

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.

 

[HeavenWind]

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)

 

[nasu]

The distance traveled is not vertical and is not due to motion with acceleration g.

 

[haruspex]

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?

 

[neilparker62]

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?

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