A machine pulls a 40 kg trunk 2.0 m up a ramp at a constant velocity, with the machine’s force on the trunk directed parallel to the ramp. The coefficient of kinetic friction between the trunk and the ramp is 0.40 . What are (a) the work done on the trunk by the machine’s force and (b) the increase in thermal energy of the trunk and the ramp?
The mass of the trunk is, m = 40 kg
The distance by which the trunk is pulled up, x = 2.0 m
The angle of inclination is,
The coefficient of kinetic friction is,
Using the equation for the net force, we can find the force due to the machine. From this force and displacement, we can find the work done by the machine. Using the frictional force, we can find the thermal energy generated.
Force due to Newton’s second law, F = ma (1)
The work done by the body, W = Fdcos (2)
We can resolve the gravitational force into its component along the incline and normal to the incline, we have and, respectively.
Using these components in equation (2), we get that
The net acceleration is zero, so we can write the force as:
Thus, work done by the machine can be given as:
Hence, the value of the work done is 744J.
The force due to friction is given as:
So the thermal energy generated is equal to the work done by the frictional force, which is given using equation (2) as:
Hence, the value of the increase in thermal energy is 240.2 J .
A cookie jar is moving up an incline. At a point 55 cm from the bottom of the incline (measured along the incline), the jar has a speed of 1.4 m/s . The coefficient of kinetic friction between jar and incline is 0.15 . (a) How much farther up the incline will the jar move? (b) How fast will it be going when it has slid back to the bottom of the incline? (c) Do the answers to (a) and (b) increase, decrease, or remain the same if we decrease the coefficient of kinetic friction (but do not change the given speed or location)?
A 2.0 kg breadbox on a frictionless incline of angle is connected, by a cord that runs over a pulley, to a light spring of spring constant k =120N/m, as shown in Figure. The box isreleased from rest when the spring is unstretched. Assume that the pulley is mass less and frictionless. (a) What is the speed of the box when it has moved 10 cm down the incline? (b) How far down the incline from its point of release does the box slide before momentarily stopping, and what are the (c) magnitude (d) direction (Up or down the incline) of the box’s acceleration at the instant the box momentarily stops?
In Fig. 8-46, a spring with k=170 N/m is at the top of a frictionless incline of angle . The lower end of the incline is distance D = 1.00 m from the end of the spring, which is at its relaxed length. A 2.00 kg canister is pushed against the spring until the spring is compressed 0.200 m and released from rest. (a) What is the speed of the canister at the instant the spring returns to its relaxed length (which is when the canister loses contact with the spring)? (b) What is the speed of the canister when it reaches the lower end of the incline?
A 0.42 kg shuffleboard disk is initially at rest when a player uses a cue to increase its speed to 4.2 m/s at constant acceleration. The acceleration takes place over a 2.0 m distance, at the end of which the cue loses contact with the disk. Then the disk slides an additional 12 m before stopping. Assume that the shuffleboard court is level and that the force of friction on the disk is constant. What is the increase in the thermal energy of the disk–court system (a) for that additional 12 m and (b) for the entire 14 m distance? (c) How much work is done on the disk by the cue?
A 3.2 kg sloth hangs 3.0 m above the ground. (a) What is the gravitational potential energy of the sloth-Earth system if we take the reference point y=0 to be at the ground? If the sloth drops to the ground and air drag on it is assumed to be negligible, what are the (b) kinetic energy and (c) speed of the sloth just before it reaches the ground?
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