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3. Suppose you have a box of mass m on a horizontal surface. It is connected by a rope to a bucket of mass M that is hanging off a ledge via a pulley system. Suppose the box and the horizontal surface have coefficient of static friction s. (a) Suppose the box does not move. What must µ‹ be, at least? (b) Suppose that μg is twice the value you found in part (a). In addition, water is being added to the bucket at a rate of rw (mass/time) until the box moves. What is the total mass of the water Mw when the box begins to move? How long, T, does this take? (c) Suppose μ = μs/3. What is the acceleration of the box after it starts moving?

3. Suppose you have a box of mass m on a horizontal surface. It is connected by a rope to a bucket of mass M that is hanging off a ledge via a pulley system. Suppose the box and the horizontal surface have coefficient of static friction s. (a) Suppose the box does not move. What must µ‹ be, at least? (b) Suppose that μg is twice the value you found in part (a). In addition, water is being added to the bucket at a rate of rw (mass/time) until the box moves. What is the total mass of the water Mw when the box begins to move? How long, T, does this take? (c) Suppose μ = μs/3. What is the acceleration of the box after it starts moving?

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**Problem 3: Engineering Dynamics** Suppose you have a box of mass \(m\) on a horizontal surface. It is connected by a rope to a bucket of mass \(M\) that is hanging off a ledge via a pulley system. The box and the horizontal surface have a coefficient of static friction \(\mu_s\). 1. <b>Figure Description:</b> - Box of mass \(m\) on a horizontal plane. - Bucket of mass \(M\) hanging off a ledge. - Pulley system connecting the box and the bucket. - Coefficient of static friction \(\mu_s\). Detailed sections: - **(a)** Suppose the box does not move. What must \(\mu_s\) be, at least? - **(b)** Suppose that \(\mu_s\) is twice the value you found in part (a). In addition, water is being added to the bucket at a rate of \(r_w\) (mass/time) until the box moves. What is the total mass of the water \(M_w\) when the box begins to move? How long, \(T\), does this take? - **(c)** Suppose \(\mu_k = \mu_s/3\). What is the acceleration of the box after it starts moving? **Graph Explanation:** - **Static Friction Analysis:** The static friction force \(f_s\) must counteract the pulling force exerted by the bucket. \[ f_s \geq M \cdot g \] where \(g\) is the acceleration due to gravity. - **Water Addition Analysis:** As water is added to the bucket, the mass increases according to: \[ M_w = r_w \cdot T \] The box will move when the force due to the combination of the bucket and the added water overcomes the static friction. - **Kinetic Friction and Acceleration:** Once the box moves, the friction changes to kinetic friction, given by: \[ \mu_k = \frac{\mu_s}{3} \] The net force \(F_{net}\) on the system will determine the acceleration \(a\): \[ F_{net} = (M + M_w) \cdot g - \mu_k \cdot m \cdot g = (m \cd
Transcribed Image Text:**Problem 3: Engineering Dynamics** Suppose you have a box of mass \(m\) on a horizontal surface. It is connected by a rope to a bucket of mass \(M\) that is hanging off a ledge via a pulley system. The box and the horizontal surface have a coefficient of static friction \(\mu_s\). 1. <b>Figure Description:</b> - Box of mass \(m\) on a horizontal plane. - Bucket of mass \(M\) hanging off a ledge. - Pulley system connecting the box and the bucket. - Coefficient of static friction \(\mu_s\). Detailed sections: - **(a)** Suppose the box does not move. What must \(\mu_s\) be, at least? - **(b)** Suppose that \(\mu_s\) is twice the value you found in part (a). In addition, water is being added to the bucket at a rate of \(r_w\) (mass/time) until the box moves. What is the total mass of the water \(M_w\) when the box begins to move? How long, \(T\), does this take? - **(c)** Suppose \(\mu_k = \mu_s/3\). What is the acceleration of the box after it starts moving? **Graph Explanation:** - **Static Friction Analysis:** The static friction force \(f_s\) must counteract the pulling force exerted by the bucket. \[ f_s \geq M \cdot g \] where \(g\) is the acceleration due to gravity. - **Water Addition Analysis:** As water is added to the bucket, the mass increases according to: \[ M_w = r_w \cdot T \] The box will move when the force due to the combination of the bucket and the added water overcomes the static friction. - **Kinetic Friction and Acceleration:** Once the box moves, the friction changes to kinetic friction, given by: \[ \mu_k = \frac{\mu_s}{3} \] The net force \(F_{net}\) on the system will determine the acceleration \(a\): \[ F_{net} = (M + M_w) \cdot g - \mu_k \cdot m \cdot g = (m \cd
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